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OF DYNAMICS RATIONAL SEMIGROUPS DAVID BOYD AND RICH STANKEWITZ Contents 1. Introduction 1.1. The expanding property of the Julia set 2. Uniformly Perfect Sets 2.1. Logarithmic capacity 2.2. Julia sets of nitely generated rational semigroups are uniformly perfect 3. Nearly Abelian Semigroups 3.1. Wandering domains 3.2. Stable Domains 3.3. Some properties of Stable Basins 3.4. Examples 4. Completely invariant Julia sets 4.1. Components of W(G) 4.2. Polynomial semigroups 5. An Invariant measure for nitely generated rational semigroups 5.1. Discussion of the inequalities (5.2) 5.2. Discussion and Proof of Corollary 5.1 5.3. Proof of Theorem 5.2 5.4. Proof of the weak convergence of a . n 5.5. Relationships between and T m 5.6. Convergence of T m 5.7. The existence and regularity of 5.8. Proof of Lemma 5.2 5.9. Proof of the inequalities (5.2) 5.10. Proof of the inequality (5.3) 5.11. The Support of 6. The Filled-in Julia Set for Polynomial Semigroups of Finite Type 6.1. Polynomial Semigroups of Finite Type 6.2. Relationship Between Critical Points and K(G) 6.3. Alternative De nitions for K(G) 1 2 4 7 9 11 13 16 18 21 22 23 27 34 41 44 45 47 48 49 50 51 51 56 58 59 59 60 66 69 2 DAVID BOYD AND RICH STANKEWITZ 7. Ahlfors Theory of Covering Surfaces References 71 75 1. Introduction These notes are based on a series of lectures given by the authors at Georg-August-Universit t in G ttingen in June 22 - July 2, 1998. a o The authors would like to thank Professors Manfred Denker and Hartje Kriete for their hospitality. For a treatment of the classical iteration theory one may see [5] and [33]. We use these texts as the basic references for such material. The material in Sections 1 3 on rational semigroup dynamics is largely taken from the papers [15] and [16] by Aimo Hinkkanen and Gaven Martin. The material in Section 4 is taken from the papers [32], [30] and [31] by Rich Stankewitz. The material from Sections 5 and 6 is taken from the papers [8], [7] and [6] by David Boyd. The study of the dynamics of rational semigroups is a generalization of the study of the dynamics associated with the iteration of a rational function de ned on the Riemann sphere. A main focus of this study is to see how far and in what sense does the classical theory of Fatou and Julia extend to this new setting. In particular, it is of interest to understand to what extent such main results as Sullivan s no-wandering-domains theorem and the Classi cation of xed components theorem hold in this more general setting. We are also interested in learning what new phenomena can occur and what new insight this might lend to the classical theory. In what follows all notions of convergence will be with respect to the spherical metric d on the Riemann sphere C. A rational semigroup G is a semigroup of rational functions of degree greater than or equal to two de ned on the Riemann sphere C with the semigroup operation being functional composition. When a semigroup G is generated by the functions {f1 , f2 , . . . , fn , . . . }, we write this as (1.1) G = f 1 , f2 , . . . , f n , . . . . In [15], p. 360 the de nitions of the set of normality, often called the Fatou set, and the Julia set of a rational semigroup are as follows: De nition 1.1. For a rational semigroup G we de ne the set of normality of G, N (G), by N (G) = {z C : a neighborhood of z on which G is a normal family} DYNAMICS OF RATIONAL SEMIGROUPS 3 and de ne the Julia set of G, J(G), by J(G) = C \ N (G). Clearly from these de nitions we see that N (G) is an open set and therefore its complement J(G) is a compact set. These de nitions generalize the case of iteration of a single rational function and we write N ( h ) = N (h) = Nh and J( h ) = J(h) = Jh . Note that J(G) J(g) and N (G) N (g) for all g G. De nition 1.2. If h is a map of a set Y into itself, a subset X of Y is: ii) backward invariant under h if h 1 (X) X; i) f orward invariant under h if h(X) X; iii) completely invariant under h if h(X) X and h 1 (X) X. It is well known that the set of normality of h and the Julia set of h are completely invariant under h (see [5], p. 54), in fact, (1.2) h(N (h)) = N (h) = h 1 (N (h)) and h(J(h)) = J(h) = h 1 (J(h)). Theorem 1.1 (Montel s Theorem). The family of all analytic maps f from a domain to C \ {0, 1, } is normal in . By using Montel s Theorem one can obtain the following result. Property 1.1. The set J(h) is the smallest closed completely invariant (under h) set which contains three or more points (see [5], p. 67). In fact, this may be chosen as an alternate de nition to the de nition of J(h) given in De nition 1.1. Proposition 1.1 ([15], p. 360). The set N (G) is forward invariant under each element of G and J(G) is backward invariant under each element of G. Proof. We make use of the fact that a family of continuous functions de ned on a domain of the Riemann sphere is equicontinuous if, and only if, it is a normal family. Let g G and z N (G). For > 0 there exists a neighborhood of z such that diamf ( ) < for all f G. Hence diam h(g( )) = diam(h g)( ) < for all h G. Hence G is equicontinuous on g( ), and we conclude that g(N (G)) N (G). Remark 1.1. Since J(G) is backward invariant we can characterize J(G) as the smallest closed set that contains three or more points which is backward invariant under each element of G. This follows since the complement of such a set is forward invariant under each element of G and therefore in the set of normality of G by Montel s theorem. 4 DAVID BOYD AND RICH STANKEWITZ Proposition 1.2 ([34], Lemma 1.1.4). If G = g1 , . . . , gN , then J(G) = 1 1 N gi (J(G)) and N (G) = N gi (N (G)). i=1 i=1 Proof. By Proposition 1.1 we have 1 N (G) N gi (N (G)). i=1 1 Take any z0 N gi (N (G)) and set wj = gj (z0 ) N (G). For any i=1 > 0 there is a neighborhood j of wj for each j = 1, . . . , N such that if f G, then diamf ( j ) < for each j = 1, . . . , N . Consider 1 the neighborhood = N gj ( j ) of z0 and note that for any f G j=1 we have diam(f gj )( ) = diamf (gj ( )) diamf ( j ) < for each j = 1, . . . , N . Hence G gj = {h gj : h G} is equicontinuous at z0 . N Since G = ( N G gj ) ( i=1 gj ) we see that G is equicontinuous at i=1 z0 . The corresponding statement for J(G) readily follows. The sets N (G) and J(G) are, however, not necessarily completely invariant under the elements of G. This is in contrast to the case of single function dynamics as noted in (1.2). Example 1.1. Let a C, |a| > 1 and G = z 2 , z 2 /a . One can easily show that J(G) = {z : 1 |z| |a|} (see [15], p. 360). Note that J(z 2 ) = {z : |z| = 1} and J(z 2 /a) = {z : |z| = |a|}. Clearly in this example J(G) is not completely invariant. We will study completely invariant Julia sets for rational semigroups in Section 4. Note also that J(G) has nonempty interior and yet J(G) = C. This is not possible for the Julia set of a single rational function. 1.1. The expanding property of the Julia set. Let G be a rational semigroup. A point z C is called exceptional if its backward orbit O (z) = {w : g G such that g(w) = z} is nite. The set of exceptional points is denoted by E(G). When G = f , we denote the set of exceptional points by E(f ). For any rational function f of degree at least 2, it is well known that |E(f )| 2 where |E(f )| denotes the cardinality of the set E(f ) (see [5], Theorem 4.1.2). If |E(f )| = 1, then f is conjugate to a polynomial. If |E(f )| = 2, then f is conjugate to a map of the form z z d where d Z and |d| 2. For general semigroups of rational functions, we have the following proposition. Proposition 1.3 ([15], Lemma 3.4). Let G be a rational semigroup. Then |E(G)| 2. If |E(G)| = 1, then G is conjugate to a polynomial semigroup. If |E(G)| = 2, then G is conjugate to a semigroup whose elements are all of the form az n for a C and n Z. DYNAMICS OF RATIONAL SEMIGROUPS 5 Remark 1.2. If G is a nitely generated semigroup, then E(G) N (G). This need not be the case if G is not nitely generated. We leave it to the reader to provide an example of such a semigroup. Proposition 1.4 ([15], Lemma 3.2). Let G be rational semigroup and let a C \ E(G). Then J(G) is a subset of the accumulations points of O (a). Let G be a rational semigroup and select an element g G. Note that J(g) J(G). We will now show how J(G) can be built up from J(g). For a collection of sets A, and a function h, we denote new collections of sets by h(A) = {h(A) : A A} and h 1 (A) = {h 1 (A) : A A}. Consider the countable collection of sets F0 = {J(g)}, Fn+1 = and F = f G f 1 (Fn ), n=0 Fn . Since J(G) is backward invariant under each f G, closed, and contains J(g), we have J(G) A F A. Lemma 1.1 ([32], Lemma 3). We have J(G) = A F A. Proof. Since the set on the right is closed, backward invariant under each f in G (since rational functions are continuous open maps) and clearly contains more that three points, it must contain J(G) as the complement is then in the set of normality of G. Remark 1.3. In fact, if we had let F0 = {{a, b, c}} where a, b, c are three points known to be in J(G) (for example, if a, b, c J(g)) and we de ned each Fn and F as above in terms of this new collection F0 , then we would arrive at the same description of J(G) as given in Lemma 1.1. This is due to the minimality condition for Julia sets as noted in Property 1.1. Corollary 1.1 ([15], Lemma 3.1). The set J(G) is perfect. Proof. Since J(g) is perfect (see [5], p. 68) and backward and forward images of perfect sets under rational maps are perfect, we see that each set in E is perfect by a routine inductive argument. The corollary then follows since the closure of a union of perfect sets is perfect. 6 DAVID BOYD AND RICH STANKEWITZ The above proof due to Rich Stankewitz (see [32]) is given as an alternative to the original proof found in [15]. Theorem 1.2 ([15], Theorem 3.1 and Corollary 3.1). If G is a rational semigroup, then the repelling xed points of the elements of G are dense in J(G). Hence also J(G) = g G J(g). Proof. The proof will follow along the lines of that of Baker in [2]. As the repelling xed points of any element g G are in J(g) and each J(g) J(G), we have that the repelling xed points of the elements of G are in J(G). We will now show that such points are dense in J(G). Pick z0 J(G) and let U be a neighborhood of z0 . We will show that U contains a repelling point of some element of G. Since J(G) is perfect, we may nd disks Bj = {z : |z aj | < } U \ (E(G) {z0 }) with disjoint closures, centered at nite points aj J(G), for 1 j 5. We denote the spherical derivative of a meromorphic function by f # ; thus f # (z) = |f (z)|/(1 + |f (z)|2 ), with the usual modi cation if z = or f (z) = . Let C be the positive constant associated with the set {Bj : 1 j 5} by the Ahlfors Five Island Theorem (see Theorem 7.4 below). Thus C is chosen so that if f is any meromorphic function de ned on the unit disk with f # (0) > C, then the unit disk contains a simply connected subdomain that is mapped conformally by f onto some Bj . If 1 j 5, then G is not normal in any neighborhood of aj . Thus by Marty s criterion (see [28], p. 75), there is some fj G and a point bj Dj = {z : |z aj | < /3} such that f # (bj ) > 3C/ . Write Ej = {z : |z bj | < /3} Bj . Then gj (z) = fj (bj +z) is meromorphic in the disc # centered at the origin of radius /3 with gj (0) = fj# (bj ) > 3C/ . Hence we deduce that gj maps some simply connected subdomain of the disc centered at the origin of radius /3 conformally onto some Bi , where 1 i 5. Thus the corresponding fj maps some simply connected subdomain of Ej , and consequently some simply connected relatively compact subdomain of Bj , conformally onto some Bi . Repeating this argument at most ve times, we nd some k with 1 k 5, and an element g G arising as a composition of the fj , such that g maps some simply connected relatively compact subdomain of Bk conformally onto Bk . It now follows that some branch of g 1 has an attracting xed point, and hence g has a repelling xed point in Bk U. The iterates of a single rational function f expand open sets which meet J(f ) as explained in the following proposition. DYNAMICS OF RATIONAL SEMIGROUPS 7 Proposition 1.5 ([5], Theorem 6.9.4). Let f be a rational function with deg f 2, let W be a non-empty open set intersecting J(f ), and let K be a compact subset of C \ E(f ). Then there exists an integer N such that K f n (W ) for all n N . Correspondingly, there is an expanding property for nitely generated semigroups of rational functions. For a rational semigroup G = f1 , . . . , fk we de ne the length of a word g = fin fin 1 fi1 with ij {1, . . . , k} to be l(g) = n. We note that it is possible for an element of G to be represented by multiple words. Proposition 1.6 ([7], Lemma 1). Let G = f1 , . . . , fk be a nitely generated rational semigroup with deg fj 2 for j = 1, . . . , k, let W be a non-empty open set intersecting J(G), and let K be a compact subset of C \ E(G). Then there exists a positive integer N such that for all n N, K g(W ) l(g)=n where g ranges over the words of G of length n. We leave the proof of Proposition 1.6 as an exercise. 2. Uniformly Perfect Sets In this section we show that J(G) is uniformly perfect when G is nitely generated. Uniformly perfect sets were introduced by A. F. Beardon and Ch. Pommerenke in 1979 in [3]. We begin with some preliminary de nitions. De nition 2.1. A conformal annulus is an open subset A of C that can be conformally mapped onto the genuine annulus Ann(0; r1 , r2 ) = {z : 0 r1 < |z| < r2 } and the modulus of such a conformal annulus is given by 1 r2 mod(A) = log . 2 r1 We note that mod(A) is a conformal invariant. De nition 2.2. A conformal annulus A is said to separate a set F if F intersects both components of C \ A and F A = . De nition 2.3. A closed curve is said to separate a set F if F intersects more than one component of C \ and F = . De nition 2.4. A closed curve is said to separate the nonempty sets A and B if there does not exist a component of C \ that intersects both A and B and if is disjoint from both A and B. 8 DAVID BOYD AND RICH STANKEWITZ Remark 2.1. Uniformly perfect sets are necessarily perfect (see [27], p. 192). Remark 2.2. By a scaling and normal families argument one can show that conformal annuli of large modulus contain genuine annuli of large modulus. Thus the compact set E is uniformly perfect if, and only if, there is a c > 0 such that for any nite z0 E and r > 0 (and r < r0 when E), the Euclidean annulus {z : cr < |z z0 | < r} meets E. / Remark 2.3. For a hyperbolic domain U C it is known (from estimates when U = C \ {0, 1}) that the hyperbolic density U (z) + as z tends to any nite point on the boundary of U . Lemma 2.1. Thus the boundary of a domain D is uniformly perfect if, and only if, there is a positive constant such that every Jordan curve in D separating D has hyperbolic length at least , with respect to the hyperbolic metric in D. Proof. We observe in the annulus of radii 1 and R > 1, the circle of radius R has hyperbolic length 2 2 / log R. This can be calculated using the density for the annulus in [23], p. 12. Hence, if D is not uniformly perfect then there exist separating annuli An of modulus 1 log Rn . The circle centered at the center of An of radius Rn 2 therefore has hyperbolic length less than or equal to 2 2 / log Rn 0. (Note that the hyperbolic density in D is less than the hyperbolic density in An .) Suppose that D is uniformly perfect. We may also assume that D since the property of being uniformly perfect is invariant under M bius maps (see [27], p. 192). Since D is uniformly perfect o there exists a c > 0 such that c (z) > (z) where (z) denotes the hyperbolic density on D and (z) denotes the (Euclidean) distance from z to D (see [3], p. 476). Let be a curve in D that separates D. Let z D be a nite point that lies in a bounded component of C \ . Letting R denote the maximum distance from z to a point on , we see that since winds around z we must have De nition 2.5. ([27], p. 192) We say that a compact subset F C is uniformly perfect if F has at least two points and if the moduli of conformal annuli in C \ F which separate F are bounded. |dz| 2R. DYNAMICS OF RATIONAL SEMIGROUPS 9 Hence the hyperbolic length of satis es c l( ) = (z)|dz| |dz| 2c. R Claim 2.1. Let U be a domain in C such that #(C \ U ) 3 and let n be (smooth) curves in U . Then if the hyperbolic length of n tends to 0, the spherical lengths of n also tend to 0. 2.1. Logarithmic capacity. In this section we state Pommerenke s criterion in terms of logarithmic capacity for a set to be uniformly perfect. We state a few facts about logarithmic capacity, but for a more thorough treatment see [1]. De nition 2.6. For a measure on a compact set F we de ne the logarithmic potential of by p (z) = F log |z | d ( ). De nition 2.7. For a measure on a compact set F we de ne S = sup p (z). z F We note that S may be in nite, as is the case when = z0 for some z0 F . From all the measures with total measure (F ) = 1, there is one that minimizes S (see [1], p. 25). This measure is called the equilibrium measure. De nition 2.8. If we call S = min{S : is a measure of total measure (F ) = 1}, then we de ne the capacity of F by cap(F ) = e S . If S = , i.e., no can be chosen such that S is nite, then we say that F is a set of zero capacity. We note that one can show that cap({w : |w z| r}) = r. Pommerenke [27] has shown that a set E is uniformly perfect if, and only if, there exists a constant > 0 such that (2.1) cap(E {w : |w z| r}) r for all z E whenever 0 < r < diam(E). We note that if 2.1 holds for all r < r0 then 2.1 holds for 0 < r < diam(E) if is replaced by r0 . This immediately implies the following lemma. diam(E) 10 DAVID BOYD AND RICH STANKEWITZ Remark 2.4. Note that this implies that if E is uniformly perfect then each component of C \ E is regular for the Dirichlet problem. Lemma 2.2. The union of nitely many uniformly perfect sets is uniformly perfect. Lemma 2.3. If A is a uniformly perfect set and B is a compact set which does not contain A, then A \ B contains a uniformly perfect subset X. Proof. Let z be a nite point in A \ B and let > 0 be chosen such that (z, ) B = . Observe that A (z, ) is a perfect and closed set. If A (z, ) is not totally disconnected, then we may select for X any component of A (z, ) which is not a single point, for it will then be a compact and connected set with more than one point and hence uniformly perfect. If A (z, ) is totally disconnected, then one may nd a simple closed curve in (z, ) \ A which separates A (see Lemma 2.4). Letting D denote the component of C \ A which does not intersect C \ (z, ) (i.e., the inside component) we let X = D A and note that it is uniformly perfect as can be seen by using Pommerenke s criterion above and the fact that A is uniformly perfect. To see this we use the fact that X is then both open in A and compact and therefore there exists a r0 > 0 such that every point of X is at a distance at least r0 from A \ X. Lemma 2.4. If A (z, ) is (nonempty) totally disconnected, perfect and closed, then there exists a simple closed curve in (z, )\A which separates A. Proof. We see in [18], p. 100 that A (z, ) is homeomorphic to the middle third Cantor set C. Let f : A (z, ) C be a homeomorphism. Consider the open set f ( (z0 , )) A) in C where z0 A and is small enough so that (z0 , ) is a subset of (z, ) and so that A is not contained in (z0 , ). Since C contains in nitely many small copies of itself, we may nd such a copy C in f ( (z0 , ) A). Note that C is open in C. Now f 1 (C ) is open in A (z, ) and as such equals the intersection of A (z, ) with an open subset U of (z0 , ). Since C is closed in C, f 1 (C ) is closed in A (z, ) and hence no points of f 1 (C ) can approach the boundary of U . Say that all the points of f 1 (C ) are always a distance from the boundary of U . Using a grid of squares of size /4 we can construct a simple polygonal path U \ A that separates A. DYNAMICS OF RATIONAL SEMIGROUPS 11 Note that the set f 1 (C ) in the proof of Lemma 2.4 can be seen to be uniformly perfect when A is uniformly perfect once it was known that f 1 (C ) is both compact and open in A (z, ) by Pommerenke s criterion (without having to nd a curve ). 2.2. Julia sets of nitely generated rational semigroups are uniformly perfect. It is known that the Julia set of a rational function is uniformly perfect. Several proofs of this fact have been given, namely by Eremenko [11], Hinkkanen [13], and Ma and da Rocha [22]. ne We rst point out the following fact. Claim 2.2. Let be a simple closed curve in C and let f be a rational function. Let D be a component of C\ and C a component of C\f ( ). Then if f (D) C = , then C f (D). Proof. If C is a proper subset of f (D) then there would exist a point w f (D) C. This implies that there exists a point z D with f (z) = w. The point z cannot be in D else w = f (z) f (D). Hence z D and w = f (z) f ( ). Theorem 2.1 ([16], Theorem 3.1). Let G = g1 , g2 , . . . , gN be a nitely generated rational semigroup. Then the Julia set J(G) is uniformly perfect. Proof. Let J(G) denote the Julia set of G and Ji the Julia set of the generator gi . Since connected closed sets containing at least two points are uniformly perfect, we shall assume that J(G) is not connected and not uniformly perfect. In particular, then N (G) = . Since the union of nitely many uniformly perfect sets is uniformly perfect, we may assume that J(G) = N Ji . By Remark 1.1, there is h G such that i=1 N h 1 ( i=1 Ji ) N Ji . Now h 1 ( N Ji ) is uniformly perfect since each i=1 i=1 Ji is uniformly perfect and h is rational (to see this in detail one can argue as in the proof of Lemma 2 in [13]). By Lemma 2.3 we choose X to be a uniformly perfect compact subset of N h (Jj ) \ ( 1 Ji ). i=1 Note that J(G) has positive logarithmic capacity. Thus N (G) has a hyperbolic metric (which is de ned in each component of N (G) separately). Since J(G) is not uniformly perfect, there is a sequence of simple closed curves k N (G) such that each k separates J(G) and such that the hyperbolic length l( k ) 0 12 DAVID BOYD AND RICH STANKEWITZ as k . Since X is uniformly perfect, we may assume that no k separates X. Thus for each k there exists Dk a component of C \ k such that X Dk = and J(G) Dk = . For each k = 1, 2, . . . choose hk G to be an element of shortest word length such that X hk (Dk ). The existence of these maps follows from the density of the repelling xed points of the elements of G in J(G) and the use of Claim 2.2. (Of course, there may be no uniqueness in the choice for hk even if the word length is minimal.) Now each hk can be written in the form hk = gi1 gi2 gim , where m = m(k) is as small as possible and each i {1, 2, . . . N } (and each integer i depends on k). Passing to a subsequence and, if necessary, relabeling the generators, we may assume that gi1 = g1 for all k. Let us de ne fk = gi2 gi3 gim . We rst claim that there are only nitely many k for which hk ( k ) separates X from J1 . To see this, simply note that hk is an analytic map from N (G) into N (G) and therefore a contraction in the hyperbolic metric. Thus the length of hk ( k ) is less than the length of k and this is going to zero. But any curve separating X from J1 has a length which is bounded below by a xed constant since both these sets have positive diameter (see Claim 2.1). Similarly, there are only nitely many k for which fk ( k ) separates X from at least one Ji . Thus, after passing to a subsequence, we may assume that neither hk ( k ) nor fk ( k ) ever separates X from J1 . By the minimality in the word length of hk , the set fk (Dk ) does not contain X while g1 (fk (Dk )) does. Now fk ( k ) separates X for only nitely many k, because any loop that separates X has hyperbolic length (in the hyperbolic metric of C \ X) bounded below by a xed positive constant as X is uniformly perfect (and since the hyperbolic metric of C \ X is smaller than that of N (G)). Thus, after again passing to a subsequence, we may assume that fk ( k ) never separates X. Similarly, we may assume that fk ( k ) never separates J1 . Write k = fk ( k ). We have arrived at the situation where k does not separate X or J1 , nor does k separate X from J1 . Thus X and J1 lie in the same component of C \ k . This component does not meet fk (Dk ) and in particular fk (Dk ) does not meet J1 . Now hk (Dk ) = g1 (fk (Dk )) covers X and therefore must meet J1 as both X and J1 meet the same component of C \ hk ( ). This is a contradiction, as any z fk (Dk ) 1 which maps by g1 to a point in J1 must itself be in J1 since g1 (J1 ) = J1 . But this contradicts the fact that fk (Dk ) foes not meet J1 . DYNAMICS OF RATIONAL SEMIGROUPS 13 Theorem 2.2 ([16], Theorem 4.1). Let G be a rational semigroup such that J(G) is uniformly perfect. Suppose that z0 is a superattracting xed point of an element h G. Let A be the union of all the components of N (h) in which the iterates of h tend to z0 . Then either z0 N (G) or A J(G). In particular, either z0 N (G) or z0 lies in the interior of J(G). Proof. Note rst that by the forward invariance of N (G) under G, if N (G) A = , then N (G) contains points as close to z0 as we like. We may assume that z0 J(G) and that N (G) A = , for otherwise there is nothing to prove. Close to z0 we may conformally conjugate h to z z d for some d 2. Let us use the coordinates in which h is equal to z z d . In these coordinates, let V be a small disk close to z0 = 0 contained in N (G). Also hn (V ) N (G) for all n 1, and since the application of h (that is z z d ) multiplies the argument of a point in V by d, we see that for all su ciently large n, the set hn (V ) contains an annulus centered at z0 = 0. Suppose that Vn is the component of N (G) containing hn (V ). Now if, for a certain n, the set Vn contains the annulus {z : r1 < |z| < r2 }, then for any k 1, the set Vn+k dk dk contains the annulus {z : r1 < |z| < r2 }. Since {z0 } J(h) J(G). it follows that Vn+k separates J(G). The moduli of these annuli are dk dk equal to log(r2 /r1 ) = dk log(r2 /r1 ) as k . This contradicts the assumption that J(G) is uniformly perfect. Corollary 2.1 ([16], Corollary 4.1). If G is a nitely generated rational semigroup and z0 is a superattracting xed point of some element of G, then either z0 lies in (the interior of ) the Fatou set of G or in the interior of the Julia set of G. Theorem 2.3 ([16], Theorem 5.1). There exists an in nitely generated rational semigroup G (all of whose elements have degree at least two) with the property that for any positive integer N , the semigroup G contains only nitely many elements of degree at most N , such that J(G) is not uniformly perfect, and such that G contains an element g with a superattracting xed point with J(G) J(G). 3. Nearly Abelian Semigroups A natural question regarding rational semigroups is how the algebraic structure of the semigroup a ects its dynamics. If the algebraic structure is in some way simple, this may provide information about the dynamics. In this section we discuss the concept of nearly abelian semigroups as introduced in [15]. 14 DAVID BOYD AND RICH STANKEWITZ As a motivating example we consider the following lemma which is due to Julia. Lemma 3.1. Let f and g be rational functions of degree at least two that commute, i.e., f g = g f . Then J(f ) = J(g). Proof. Since g is uniformly continuous on C in the spherical metric, the family {g f n : n 1} is normal on N (f ). This is the same family as {f n g : n 1} and so {f n : n 1} is normal on the open set g(N (f )). Thus g(N (f )) N (f ). So all the g n omit J(f ) on N (f ). As the degree of f is greater than two, J(f ) contains at least three points and so it follows that the family {g n : n 1} is normal on N (f ). Hence N (f ) N (g). By symmetry we obtain N (g) N (f ). This gives N (f ) = N (g) and hence J(f ) = J(g) as desired. In particular, if the rational semigroup G is abelian, then J(g) = J(G) for every g G by Theorem 1.2. However, we are able to obtain a similar result for a more general class of rational semigroups. De nition 3.1. A rational semigroup G is nearly abelian if there is a compact family of M bius transformations = { } with the following o properties: (i) (N (G)) = N (G) for all , and (ii) for all f, g there is a such that f g = g f . Note that when G is nearly abelian, the family (G) of M bius transo formations for which f g = g f for some f, g G by assumption is precompact, i.e., any sequence of elements of (G) contains a subsequence that converges to a M bius transformation uniformly on C. o Hence we may take to be the closure of (G). We make a couple of observations that apply when (G) is precompact, although they will not be used in what follows. First, if n is any sequence from (G) and D is any disk, it cannot be the case that n converges to a constant function on D. Further, if all the n have their poles outside a xed disk larger than D, this implies a uniform upper and lower bound for | n | in D. Here is our rst result about nearly abelian semigroups. Theorem 3.1 ([15], Theorem 4.1). Let G be a nearly abelian semigroup. Then for each g G we have J(g) = J(G). Proof. Let f be a xed element of G and consider an arbitrary element g of G. Set J = J(f ) and N = C \ J. We will show that J(g) = J. Assume for a while that this is true. Recall that J = J(f ) J(G). On the other hand, for each g G, g omits J on N so that G is normal on N . Thus N N (G). It now follows that J(G) = J, as DYNAMICS OF RATIONAL SEMIGROUPS 15 claimed. (Or one could reach the same conclusion using the fact that J(G) = g G J(g). See Theorem 1.2.) We proceed to prove that if g G, then J(g) = J. For each n 1, there is n (G) with f n g = n g f n . We begin by showing that g(N (f )) N (f ). Choose some point x N (f ) and a neighborhood U of x such that U N (f ). Then g(U ) is a neighborhood of g(x). Consider a sequence of iterates f nj on g(U ). As U N (f ) we may pass to a subsequence, say f mj , in such a manner that f mj uniformly on U and where is meromorphic on U . Since g is rational, and hence uniformly continuous on C, we have g f mj g = uniformly on U . Passing to a further subsequence without changing notation, we may assume that mj uniformly on the sphere, where is a M bius transformation. Now f mj g = mj g f mj = o uniformly on U . Hence the family {f n g : n 1} is normal on U and so {f n : n 1} is normal on g(U ). Since N (f ) is the maximal open set on which {f n : n 1} is normal, we have g(U ) N (f ). Thus g(x) N (f ) and so g(N (f )) N (f ). Hence every g n omits J(f ) on N (f ), and so the iterates of g form a normal family on N (f ). This implies that N (f ) N (g). By symmetry, we obtain N (g) N (f ). Hence N (g) = N (f ) and so J(g) = J(f ) = J. We note that (since every element of G has degree at least two) the condition (N (G)) = N (G) for all may be replaced by the condition J(f ) = J(g) for all f, g G. For if this latter property holds there clearly is a set J (of cardinality at least 3) such that J = J(f ) for all f G. Thus each f G omits J in C \ J and so C \ J N (G). Since J = J(f ) J(G) we have J = J(g) = J(G) for all g G. Then applying both sides of the equation f g = g f to J we see that (J) = J and hence (N (G)) = N (G). As f, g G are arbitrary, the result holds for all such . We remark that in many cases the assumed compactness of may be redundant. It is conjectured that the M bius symmetry group of o the Julia set of a rational function is of nite order, unless the Julia o set is C or is M bius equivalent to a circle or a line segment. There are some partial results towards this conjecture in [8]. A natural question is to what extent does the converse to Theorem 3.1 hold? A. Beardon has proved the following result. Theorem 3.2 ([4], Theorem 1). If f and g are polynomials and if J(f ) = J(g), then there is a linear mapping (z) = az + b such that f g = g f and |a| = 1. 16 DAVID BOYD AND RICH STANKEWITZ Corollary 3.1 ([15], Corollary 4.1). Let F be a family of polynomials of degree at least 2, and suppose that there is a set J such that J(g) = J for all g F. Then G = F is a nearly abelian semigroup. Proof. As each g F is a polynomial, J is compact in C. We note that J(f ) = J(g) for all f, g G since the same is true for any pair of functions in F. One way to see this is as follows. It su ces to show that for any f, g F, we have J(f g) = J. It is easy to see that J(f g) J since J is backwards invariant under f and g. If J(f g) = J, then by Remark 4.6 below, there are points of J in the basin of attraction of in nity for f g. However, J is also forward invariant under f g as it is forward invariant under f and g individually. This is a contradiction and hence J(f g) = J as claimed. As J(f ) = J(g) for all f, g G, the polynomials f and g nearly commute by Theorem 3.2. We nish by observing that the family of commutators of the form z az + b, where |a| = 1, has compact closure since the numbers b must also be bounded. The following conjecture is the strongest converse to Theorem 3.1 that we can reasonably expect. Conjecture 3.1 ([15], Conjecture 4.1). Let G be a rational semigroup and suppose that for some g G we have J(g) = J(G) and that J(G) is not the image under a M bius transformation of a circle, line segment o or the Riemann sphere. Then G is nearly abelian. We remark here that the set of all rational functions that share the same Julia set J where J is M bius equivalent to the sphere a circle o or a line segment will not even nearly commute, i.e., given any two such functions f and g there need not be a M bius transformation o such that f g = g f . Further, if we restrict ourselves to such rational functions that do nearly commute, the set of commutators (G), being a subset of the symmetries of J may be so large as to not be precompact. It is also relatively easy to construct examples of rational semigroups such that J(G) = {z : |z| = 1} such that there is some g G with J(g) = J(G) yet theer is another element h G such that J(h) J(G). 3.1. Wandering domains. One of the major di erences that appear when passing from the classical iteration theory to the dynamics of rational semigroups is the existence of wandering domains. Sullivan s result precludes wandering domains in the Fatou set of a rational function. We rst need to establish what a wandering domain would mean for a rational semigroup. DYNAMICS OF RATIONAL SEMIGROUPS 17 De nition 3.2. Let G be a rational semigroup. Given a component U of N (G) and an element g G, we let Ug denote the component of N (G) containing g(U ). The component U is called a wandering domain if there are in nitely many distinct components in {Ug : g G}. We remark that g(U ) is usually a proper subset of Ug , and further there can be in nitely many distinct elements hj of G such that the sets hj (U ) are all contained in the same component of N (G), yet the sets hj (U ) might still be mutually disjoint. Hinkkanen and Martin provide an example of an in nitely generated polynomial semigroup (of nite type) that has a wandering domain. See [15], 5. They further provide an example where the wandering domain returns to the same component in nitely often. They have made the following conjecture. Conjecture 3.2 ([15], Conjecture 5.1). Let G be a nitely generated rational semigroup. Then G has no wandering domains. Some partial results have been made in this direction. One such result has to do with hyperbolic rational semigroups. A hyperbolic rational semigroup G satis es the property that J(G) is disjoint from the closed post-critical set of G. This generalization of hyperbolic rational maps was established independently by Hiroki Sumi in [34] and by Hinkkanen and Martin in [14]. Both proved that hyperbolic rational semigroups have no wandering domains. Note that these hyberbolic rational semigroups need not be nitely generated. Sumi has also established a no-wandering-domains theorem for sub- and semi-hyperbolic rational semigroups. Another no-wandering-domains type result is the following theorem. Theorem 3.3 ([15], Theorem 5.1). Let G be a nearly abelian rational semigroup. Then G has no wandering domains. Proof. Let f G be a rational map of degree at least two. Then as G is nearly abelian, we have J(f ) = J(G). Let (G) be the precompact family associated with the commutative properties of G as in De nition 3.1. For a single rational function f , the Fatou set N (f ) has a nite number of components that are periodic under f . (See [10], III, Theorem 2.7 and VI, Theorem 4.1. The sharp bounds on the number of non-repelling cycles and periodic components was found by Shishikura, see [29].) Further, by Sullivan s no-wandering-domains theorem, every component of N (f ) maps under some iterate onto a periodic component of N (f ). Thus we may replace f by a suitable iterate of f to assume that if U is a periodic component of f , then U is xed, that is, f (U ) = U . Let U be the collection of all xed components of f and 18 DAVID BOYD AND RICH STANKEWITZ let A be the set of all components of N (G) of the form (U ), where (G) and U U. It is easy to verify from the precompactness of the family (G), that A consists of a nite number of components of N (G). (For instance, we may normalize so that N (G) and then observe that there are only nitely many components whose area is larger than any given constant. Since (G) is precompact, there is a uniform bound on the amount by which any element of (G) can decrease the area of any U U.) We now observe that if g G and U U, then g(U ) A. To see this, simply observe that for every integer m, we have g(U ) = g(f m (U )) = m (f m (g(U ))), and if m is taken su ciently large, then f m (g(U )) U. Next let V be any component of N (G) and suppose that V is wandering. Choose an in nite sequence gi G such that the sets gi (V ) = Vi are disjoint. Choose an integer n such that f n (V ) = U U. As f n has nite degree, the collection {f n (Vi )} must contain an in nite number of i=1 components of N (G). However, for each i we see that for some i (G). However, it is again easy to see from the precompactness of the set (G) that in fact the set { (A) : (G)} is a nite collection of components of N (G), which yields the desired contradiction. 3.2. Stable Domains. Sullivan s no-wandering-domains theorem together with the classi cation of the periodic components of the Fatou set of a rational function describe the stable dynamics of an arbitrary rational function of degree at least two. In this section we present a partial classi cation of the dynamics of a rational semigroup on a stable domain. See [15], pp. 362, 374 379. De nition 3.3. Recall that we write Ug for the component of N (G) containing g(U ). We de ne the stabilizer of U to be If GU contains an element of degree two or more, we shall say that U is a stable basin for G. Clearly, GU is a subsemigroup of G. In particular, N (G) N (GU ), where the containment may be strict. De nition 3.4. Given a stable basin U for G we say that it is (i) attracting if U is a subdomain of an attracting basin of each g GU with deg g 2; GU = {g G : Ug = U }. f n (Vi ) = f n (gi (V )) = ( i gi f n )(V ) = i (gi (U )) i (A), DYNAMICS OF RATIONAL SEMIGROUPS 19 Remark 3.1. This classi cation is not exhaustive. See Example 3.4 below. However, we will show in Theorem 3.6 that De nition 3.4 is a complete classi cation for nearly abelian semigroups. Before we discuss these de nitions, we introduce a concept which in part generalizes to semigroups the relationship between the dynamics of a rational function and the dynamics of an iterate of the function. (ii) superattracting if U is a subdomain of a superattracting basin of each g GU with deg g 2; (iii) parabolic if U is a subdomain of a parabolic basin of each g GU with deg g 2; (iv) Siegel if U is a subdomain of a Siegel disk of each g GU with deg g 2; (v) Herman if U is a subdomain of a Herman ring of each g GU with deg g 2. De nition 3.5. A subsemigroup H of a semigroup G is said to be of nite index if there is a nite collection of elements {g1 , g2 , . . . , gn } of G {Id} such that G = g1 H g2 H gn H. If n is chosen to be as small as possible, we say that H has index n in G. For instance the subsemigroup H of a nitely generated semigroup G consisting of all words of length some multiple of an integer n has nite index in G. (As, for instance, the words of even length in G.) Thus f 2 , g 2 , f g, gf has index 3 in f, g : we may take g1 = Id, g2 = f, g3 = g. De nition 3.6. We say that a subsemigroup H of G has co nite index or nite coindex if there is a nite collection of elements g1 , g2 , . . . , gn of G {Id} such that for every g G there is j {1, 2, . . . , n} such that gj g H. The coindex of of H in G is the smallest such number n. If the semigroup were a group the two notions would coincide. In the example above, the subsemigroup f 2 , g 2 , f g, gf has coindex 2 as well as index 3. For the coindex, note that we may take g1 = f and g2 = Id. Theorem 3.4 ([15], Theorem 2.4). If H is a nite index or nite coindex subsemigroup of G, then N (H) = N (G) and J(H) = J(G). 20 DAVID BOYD AND RICH STANKEWITZ Proof. It su ces to show that N (H) = N (G), for then it immediately follows that also J(H) = J(G). Since H is a subsemigroup of G, we have N (G) N (H). It remains to be proved that N (H) N (G). Suppose that H is a nite index subsemigroup of G. If fj is a sequence of elements of G, we may pass to a subsequence without changing notation and assume that each fj can be written as fj = g hj where hj H and g {g1 , . . . , gn }, where g is independent of j and where the set {g1 , . . . , gn } is as in De nition 3.5. If U is a domain with U N (H) then we may pass to a further subsequence and assume that hj uniformly in U . Hence fj g uniformly in U . It follows that U N (G) and hence N (H) N (G). Suppose now that H is a co nite index subsemigroup of G, and let {g1 , . . . , gn } be as in De nition 3.6. If fj is a sequence of elements of G, we may pass to a subsequence without changing notation and assume that hj = g fj H where g is xed with g {g1 , . . . , gn }. Suppose that z0 N (H). Let U be a spherical disk with center z0 and with U N (H). We may pass to a further subsequence and assume that hj uniformly in U . Shrink U , if necessary, so that (V ) omits some non-empty open disk D(U ) in C. Let D (U ) be a non-empty open disk whose closure is contained in D(U ). Then the functions fj eventually omit the set g 1 (D (U )) in U , so the functions fj form a normal family in U . It follows that G is normal in U , and hence N (H) N (G), as desired. Remark 3.2. This theorem generalizes the well known fact that for any rational function f of degree at least two and for any integer n 1, we have that J(f ) = J(f n ). Example 3.1. Let h be a polynomial of degree at least two with distinct components A and B of N (h) such that h(A) = h(B) = A and A contains the (super)attracting xed point of h. Let g be a polynomial of degree at least two with distinct components U and V of N (g) such that g(U ) = g(V ) = U , U B, V A, and V . There is an integer m 1 such that hm (V ) V and hm (U ) V . Set f = hm and G = f, g . Hence U and V are components of N (G). It is easy to see that GV = {f F : F G}. Thus GV is of coindex 1 in G, while GV is not of nite index in G since g n f F G for all F G and n 1. Furthermore, GV is not nitely generated even if G is. For if GV = g1 , . . . , gk , then gi G for all i so that gi = f Fi where Fi G. But f g n GV for all n 1, and not every f g n can lie in g1 , . . . gn . DYNAMICS OF RATIONAL SEMIGROUPS 21 Theorem 3.5. Let G be a rational semigroup with no wandering domains. Let U be any component of the Fatou set. Then the forward orbit of U under G, that is, {Ug : g G}, contains a stable basin of co nite index, i.e., a stable basin W such that GW has co nite index in G. Proof. Let G and U be described as above. Since U is not a wandering domain, the forward orbit of U is nite, where we always include the domain itself in its forward orbit even if Id G. Label the components / of the forward orbit U1 , U2 , . . . , Um , with U1 = U . If for every j there is a gj G such that gj (Uj ) U1 , then GU1 is easily seen to have co nite index in G. (Namely, if g G and g(U1 ) Uj , then (gj g)(U1 ) U1 and hence gj g GU1 .) Otherwise choose k 2 such that U1 does not lie in the forward orbit of V = Uk . The forward orbit of V is then contained in {U2 , . . . , Um }, so that the number of components in the forward orbit of V is strictly less than that of U . Proceeding by the obvious induction we nd a component W whose forward orbit has fewest components, and then W = Ui for some i with 1 i m. Then for every h G, for the component Wh of the forward orbit of W there is a function g belonging to a xed nite subset of G, such that g (Wh ) W . Thus Wg h = W so that g h GW and it follows that GW has co nite index in G. Corollary 3.2. Let G be a nearly abelian rational semigroup. Let U be any component of the Fatou set. Then the forward orbit of U under G, {Ug : g G}, contains a stable basin of co nite index. 3.3. Some properties of Stable Basins. We next discuss a few simple features of some stable basins for rational semigroups. First we point out that a stable basin can be attracting for a semigroup G, and yet, there need not be a common attracting cycle xed by each g G. For instance let f (z) = z 2 + c and g(z) = z 2 + d where c, d C. If |c|, |d| are su ciently small, then the disk D(1/2) of radius 1/2 centered at 0 is mapped into the disk D(1/4) by f n , g n for some large n. Thus G = f n , g n is a polynomial semigroup which contains {z : |z| < 1/2} in its Fatou set. This disk contains the attracting cycles for f and g (and hence for f n and g n ) and these are di erent if c = d. Every h G maps D(1/2) into D(1/4) and thus contains a (super)attracting xed point for h. (Question: Is is possible to show that no h G is superattracting?) In the case when G is nearly abelian, we have the following theorem, whose proof can be found in [15], pp. 376 378. 22 DAVID BOYD AND RICH STANKEWITZ Theorem 3.6 ([15], Theorem 6.2). Let G be a nearly abelian rational semigroup and U a stable component of N (G). Then U is either attracting, superattracting, parabolic, Siegel or Herman (in the sense of De nition 3.4). In the Siegel case, the basin U contains a single cycle xed by each element of GU . If U is of Siegel or Herman type, then GU is abelian. 3.4. Examples. Example 3.2 (Common parabolic basins). Set f (z) = z 2 3/4 and g = f . Then f has a parabolic cycle at z = 1/2. Note that if (z) = z, then f = f and 2 (z) = z, so that g is actually a conjugate of f and so has a parabolic cycle at 1/2. The semigroup G = f, g is nearly abelian since f g = f f = f 2 = 2 f 2 = f f = g f. (This also follows from Corollary 3.1.) Thus a nearly abelian semigroup can have di erent parabolic cycles in the same stable basin. More precisely, there is a component U of N (G) containing the origin such that each of f 2 and g 2 maps U onto itself and has a parabolic xed point on U , the xed point being 1/2 for f 2 and 1/2 for g 2 . Example 3.3 (Common superattracting basins). Set f (z) = (z 2 c2 )2 +c and g = c (z 2 c2 )2 , i.e., if (z) = z then g = f . Then c is a superattracting xed point for f and c is a superattracting xed point for g. As before, we can see that f, g is a nearly abelian polynomial semigroup. If |c| is small enough, then both f and g map the disk {z : |z| < 1/2} into itself, and thus f and g have a common superattracting basin. Example 3.4 (Mixed Basin). Set f (z) = z/(1+z z 2 ) and g(z) = z+z 2 where 0 < < 1. Then J(g) is a Jordan curve (see [5], Theorem 9.9.3) while J(f ) is a Cantor subset of the real line (can be shown using the fact that 1/f (1/z) = z + 1 1/z). The mapping f has a parabolic xed point at 0, N (f ) is connected, each of the upper and lower half planes is completely invariant under f , and there is > 0 such that the interval (0, ) N (f ) N (g) because f ((0, )) (0, ) and g((0, )) (0, ). Let G = f, g . (Note that G is not nearly abelian!) Then each h G has an attracting or parabolic xed point at 0. If is small enough and we set B = {z : |z | < }, then f (B) B (to see this note that 1/f (1/z) = z + 1 1/z). We claim that g(B) B. We will leave the details as an exercise, but we will remark that it su ces to choose any such that 0 < < (1 )/2. It follows that B N (G) = and therefore that 0 N (G). Thus the stable basin for G containing B is DYNAMICS OF RATIONAL SEMIGROUPS 23 contained in a parabolic basin for f and contained in an attracting basin for g and with the parabolic/attracting xed point in its boundary. 4. Completely invariant Julia sets The material from this section is taken from [32] and [30]. We have seen earlier that the Julia set J(G) of a rational semigroup G need not be completely invariant under all the elements of G (see Example 1.1). This is in contrast to the classical situation where J(f ) is completely invariant under each iterate f n . The question then arises, what if we required the Julia set of the semigroup G to be completely invariant under each element of G? That is, what if we extended the de nition of a Julia set given in Property 1.1? We will consider in this section some of the consequences of such an extension which is given in the following de nition. De nition 4.1. For a rational semigroup G we de ne the completely invariant Julia set I = I(G) = {S : S is closed, completely invariant under each g G, #(S) 3} where #(S) denotes the cardinality of S. We note that I(G) exists, is closed, is completely invariant under each element of G and contains the Julia set of each element of G by Property 1.1. De nition 4.2. For a rational semigroup G we de ne the completely invariant set of normality of G, W = W (G), to be the complement of I(G), i.e., W (G) = C \ I(G). Note that W (G) is open and it is also completely invariant under each element of G. So we see that we that in the e ort to generalize the dynamics associated with the iteration of a rational function to the more general dynamics of rational semigroups, we are able to extend certain key notions in more than one way. In particular, we can de ne our Julia set in terms of normality, as we did in de ning J(G) or in terms of complete invariance, as we did in de ning I(G). It is of interest to pursue a greater understanding of how these two extensions di er, and to learn which is better for studying certain phenomena. One key di erence in the theory is that when studying the action of the elements of the semigroup, one nds that components of the set of normality N (G) only map into other components and not onto as in the action of the elements on the components of the completely invariant set of normality W (G) (see Lemma 4.4). This, of course, has a 24 DAVID BOYD AND RICH STANKEWITZ large impact on how one works to extend, and even de ne, the concepts involved in two cornerstone theorems of iteration theory, the classi cation of the xed components and Sullivan s no-wandering-domains theorem. We will see below that the extension of the Julia set given by J(G) is better if one wishes to study the dynamics on the extension of the set of normality. This is seen in Theorem 4.2 which states that if a semigroup G is generated by two polynomials with unequal Julia sets, then I(G) must necessarily be the entire Riemann sphere C. Hence , in such a case, the completely invariant set of normality is empty and so there are no dynamics on its components of which to study. In this case, however, J(G) is a compact subset of the plane C and hence there are dynamics on the components of N (G) to be studied. (If the Julia sets of the two generators are equal, then both J(G) and I(G) are equal to this common Julia set.) We note that if one is studying dynamics from the point of view that complete invariance is required, then, of course, the extension given by I(G) is better. We now compare the sets I(G) and J(G). Example 4.1. Suppose that G = f, g and J(f ) = J(g). Then I = J(f ) = J(g) since J(f ) is completely invariant under f and J(g) is completely invariant under g. It is easily veri ed that if J(f ) = J(g), then J(G) = J(f ) = J(g). We will see in the following example, however, that it is not always the case that J(G) = I(G). Example 4.2. Let a C, |a| > 1 and G = z 2 , z 2 /a . One can easily show that J(G) = {z : 1 |z| |a|} (see [15], p. 360) while I(G) = C. Note that J(z 2 ) = {z : |z| = 1} and J(z 2 /a) = {z : |z| = |a|}. Lemma 4.1 ([32], Corollary 2). For a rational semigroup G, we have J(G) I(G). Proof. Since the W (G) is forward invariant under each element of G with complement I(G) which has more than 3 points, it must lie in the set of normality of G. Let G be a rational semigroup and select an element g G. Note that J(g) I(G). We will now show how I(G) can be built up from J(g). For a collection of sets A, and a function h, we denote new collections of sets by h(A) = {h(A) : A A} and h 1 (A) = {h 1 (A) : A A}. DYNAMICS OF RATIONAL SEMIGROUPS 25 Choose g G. Let us de ne the following countable collections of sets: E0 = {J(g)}, E1 = f G f 1 (E0 ) f 1 f (E0 ), f G En+1 = and E = f G (En ) f (En ), f G n=0 En . Since I is completely invariant under each f G and contains J(g), we have I A E A. Since I is also closed, we have (4.1) I A. A E The following lemma shows that these two sets are actually equal. Lemma 4.2 ([31], Lemma 3.2.1). We have I= A E A. Proof. We only have I A E A yet to establish. Since the set on the right is closed and contains J(g) (and therefore more than three points), it remains only to show that it is also completely invariant under each f G. We will use the fact that for a non-constant rational function h and a subset B of C we have h 1 (B) = h 1 (B) since h is a continuous open map. Using this fact we see that f 1 ( A E A) = f 1 ( A E A) = A E f 1 (A) A. A E Also, by the continuity of f , we have f( A E A) f ( A) = A E A E A E f (A) A. A E Remark 4.1. In fact, if we had let E0 = {{a, b, c}} where a, b, c are three points known to be in I(G) (for example, if a, b, c J(g)) and we de ned each En and E as above in terms of this new collection E0 , then we would arrive at the same description of I(G) as given in So we conclude that I A. 26 DAVID BOYD AND RICH STANKEWITZ Lemma 4.2. This is due to the minimality condition for Julia sets as noted in Property 1.1. For technical reasons we will, however, use the previous description of I obtained from letting E0 = {J(g)}. Corollary 4.1 ([31], Corollary 3.2.3). The set I(G) has no isolated points; i.e., I(G) is perfect. Proof. Since J(g) is perfect (see [5], p. 68) and backward and forward images of perfect sets under rational maps are perfect, we see that each set in E is perfect by a routine inductive argument. The corollary then follows since the closure of a union of perfect sets is perfect. Recall the de nition of uniformly perfect sets given in De nition 2.5. It is known that Julia sets of rational functions (see [22], [13], and [11]) and Julia sets of nitely generated rational semigroups (see [16]) are uniformly perfect. We put forth the following conjecture due to Aimo Hinkkanen. Conjecture 4.1. The set I(G) is uniformly perfect when G is nitely generated. Lemma 4.3 ([31], Lemma 3.2.5). Let B be a set which is completely invariant under each f G. If I B has nonempty interior relative to B, then I B \ {at most two points}. Proof. We will use the following elementary fact: For any sets D and C and any function h we have (4.2) By hypothesis we select an open disc such that B I and B = . By Lemma 4.2 we see then that there exists a set A in En , say, such that A = . Since A En , it can be expressed as A = hn h1 (J(g)), where each hj {f : f G} {f 1 : f G}. Considering each hj as a map on subsets of C, as opposed to a map on points of C, we can de ne the inverse maps h accordingly, i.e., j h1 = f implies h = f 1 and h2 = f 1 implies h = f . The h are not 1 2 j true inverses since f 1 (f (A)) may properly contain A. The fact (4.2) does imply, however, that (4.3) (4.4) (4.5) (4.6) (4.7) A = = hn h1 (J(g)) = hn 1 h1 (J(g)) h ( ) = n . . . = h1 (J(g)) h h ( ) = 2 n = J(g) h1 h ( ) = . n D h(C) = if and only if h 1 (D) C = . = DYNAMICS OF RATIONAL SEMIGROUPS 27 Since each h maps open sets to open sets (as each f, f 1 do) we j see that U = h h ( ) is open. We observe that by the expanding 1 n property of Julia sets (see [5], p.69 ) that we have n=1 g n (U ) = C \ E(g), where E(g) is the set of (at most two) exceptional points of g. Since the complete invariance of B and I under each of the maps f G implies that U B I, we have B \ E(g) B n=1 g n (U ) n=1 B g n (U ) = n=1 g n (B U ) E. The result then follows since I is closed. Property 4.1 ([31], Corollary 3.2.6). If I(G) has nonempty interior, then I(G) = C. Proof. Letting B = C in Lemma 4.3 gives the result. Corollary 4.2 ([31], Corollary 3.2.10). If J(G) has nonempty interior, then I = C. 4.1. Components of W(G). It is well known in iteration theory that the set of normality of a rational function can have only 0, 1, 2, or in nitely many components (see [5], p. 94). In this section we generalize this result by showing that the completely invariant set of normality of a rational semigroup can have only 0, 1, 2, or in nitely many components. The proof not only generalizes the iteration result, but it also provides an alternative proof for it. The material in this section is taken entirely from [30]. Theorem 4.1 ([30], Theorem 1). For a rational semigroup G the set W (G) can have only 0, 1, 2, or in nitely many components. Lemma 4.4 ([30], Lemma 1). If W0 is a component of W , then f (W0 ) is also a component of W for any f G. Proof. Let W1 be the component of W that contains f (W0 ). We show that f (W0 ) = W1 . Suppose to the contrary that z W1 \ f (W0 ). Since f is continuous on the compact set W0 and an open map on W0 , we have f (W0 ) f ( W0 ) f (I) I. Let be a path in W1 connecting z to a point w f (W0 ). Hence must cross f (W0 ) I. This contradicts the fact that W1 and so we conclude that f (W0 ) = W1 . Since the remainder of this section will be devoted to the proof of Theorem 4.1, we will assume that W has L components where 2 L < + . We remark here that the strategy will be to show that each of the L components of W is simply connected and then the result will follow by an application of the Riemann-Hurwitz relation. 28 DAVID BOYD AND RICH STANKEWITZ De nition 4.3. Let W have components Wj for j = 0, . . . , L 1. Remark 4.2. We see by Lemma 4.4 that each f G (and hence each f 1 as well) permutes the Wj for j = 0, . . . , L 1 since f is a continuous map of W onto W . We may assume that W0 , else we may impose this condition by conjugating each f G by the same rotation of the sphere. De nition 4.4. For j = 1, . . . , L 1, we de ne where the winding number is given by Ind (z) = (1/2 i) 1/(w z) dw. If z Kj and the simple closed curve Wj is such that Ind (z) = 1, then we say that works for z Kj . In order to properly de ne K0 we rst need to move W0 so that it no longer contains . Let be a rotation of the sphere so that (W1 ) and denote Wj = (Wj ) for j = 0, . . . , L 1. De nition 4.5. We de ne and Kj = {z Wj : there exists a simple closed curve Wj such that Ind (z) = 1} / K0 = {z W0 : there exists a simple closed curve W0 such that Ind (z) = 1} / K0 = 1 (K0 ). If z K0 and simple closed curve W0 is such that Ind ( (z)) = 1, then we say that the simple closed curve 1 ( ) works for z K0 . Remark 4.3. Note that saying 1 ( ) works for z K0 does not necessarily imply that Ind 1 ( ) (z) = 1, since it may be the case that Ind ( ( )) = 1 and hence Ind 1 ( ) (z) = 0 since z lies in the unbounded component of C \ 1 ( ). De nition 4.6. We de ne L 1 K= j=0 Kj . De nition 4.7. We de ne Wj = W j K j . Lemma 4.5 ([30], Lemma 2). For j = 0, . . . , L 1, the set Wj is open, connected and simply connected. Thus each Kj is the union of the holes in Wj . Proof. Suppose that 1 j L 1, so that Wj is a bounded domain in the complex plane. De ne A to be the unbounded component of C \ Wj . Hence B = C \ A is open, connected and simply connected. DYNAMICS OF RATIONAL SEMIGROUPS 29 Let F be a bounded component of C \ Wj . Since A and F are each components of the closed set C \ Wj , there exists a simple polygon Wj which separates A from F (see [25], p. 134). Hence we see that F Kj . Since F was an arbitrary bounded component of C \ Wj , we conclude that Kj contains all the bounded components of C \ Wj , i.e., the holes of Wj . Hence Wj B. Clearly Kj cannot contain any points of A since any simple closed path Wj which would wind around such a point would have to necessarily wind around every point of A (since A is a component of the complement of Wj ) including which cannot happen. Hence we conclude Wj = B and is therefore open, connected and simply connected. We show that (W0 ) is open, connected and simply connected using the same argument as above, and this implies that W0 is open, connected and simply connected. De nition 4.8. We de ne L 1 W= j=0 Wj . Lemma 4.6 ([30], Lemma 3). If for some distinct r, s {0, . . . , L 1}, we have Wr Ws = , then either Wr Ws or Ws Wr . In particular, if Wr Ws = for some distinct r, s {0, . . . , L 1}, then Wr Ws . Proof. Let z Wr Ws . Since Wr Ws = , we may assume that z Ks , say. Let s work for z Ks . Let I s be the component of C \ s which contains z. Note that I s \ Ws = {z : s works for z} whether or not s = 0 (see De nitions 4.4 and 4.5 and Remark 4.3). Since z Wr , we have two cases, either z Kr or z Wr . Suppose that z Kr and let r work for z Kr . As s r = (since Wr Ws = ) we see that either r I s or s I r , where I r is the component of C \ r which contains z. By switching the roles of r and s, if necessary, we assume r I s and we note that this can be done since z Kr Ks . In particular, Wr I s = . If z Wr , then we still get Wr I s = since z I s . Since Wr I s = , Wr Ws = , Wr is connected, and s Ws , we conclude that Wr I s . Hence Wr Ws since s then works for every z Wr . Since Ws is simply connected we see that Wr Ws . Lemma 4.7 ([30], Lemma 4). The boundary of W0 is a nondegenerate continuum and as such contains more than three points. Note that we have W = W K. 30 DAVID BOYD AND RICH STANKEWITZ Proof. We will rst show that W0 W1 = . The set W1 cannot contain W0 as W0 and W1 is a bounded subset of C (since W1 is a bounded subset of C). The same argument also shows that (W0 ) cannot contain (W1 ) where is as in De nition 4.5, and so we conclude that W0 cannot contain W1 . By Lemma 4.6 we conclude that W0 W1 = . Since W0 is simply connected, W0 contains a nondegenerate continuum unless W0 consists of just a single point. If W0 consists of just a single point, then W0 W0 = C, but this contradicts the fact that W0 W1 = . Lemma 4.8 ([30], Lemma 5). For each j = 0, . . . , L 1, we have J(f ) Wj for each f G. Since J(G) = f G J(f ), we have J(G) Wj for each j = 0, . . . , L 1. Proof. Since f permutes the Wj by Remark 4.2, we may select a positive integer n so that f n (Wj ) = Wj = f n (Wj ) for each j = 0, . . . , L 1. Then we have f kn (Wj ) J(f n ) = J(f ) (see [5], p. 71 and k=1 p. 51). But since f kn (Wj ) = Wj we see that Wj J(f ), since k=1 Wj J(f ) = . Lemma 4.9 ([30], Lemma 6). We have Wr Ws for distinct r, s {0, . . . , L 1}, and therefore, by Lemma 4.6, the Wj are disjoint for j = 0, . . . , L 1. Proof. If L = 2, then the proof of Lemma 4.7 shows that W0 W1 = . We assume now that L 3. We will rst show that no bounded Ws can contain any Wr with r = s. Suppose that this does occur. Then there exists a simple closed curve s Ws such that Wr I s where I s is the component of C \ s which contains the points z such that Ind s (z) = 1. Hence, by Lemma 4.8, J(G) Wr Wr I s . But since W0 C \ I s we see that J(G) W0 W0 C \ I s . This contradiction implies no bounded Ws can contain any Wr . We see that W0 cannot contain any Wr with r 1 by the following similar argument. If Wr W0 , then there exists a simple closed curve W0 such that Ind (z) = 1 for every z Wr . Let I be the component of C \ which contains Wr . So (J(G)) ( Wr ) = (Wr ) = Wr I . Since W1 C \ I (recall W1 ), we see that (J(G)) ( W1 ) = (W1 ) = W1 C \ I . This contradiction implies W0 cannot contain any Wr with r 1. Corollary 4.3 ([30], Corollary 1). The set K has no interior and therefore each Kj Wj . DYNAMICS OF RATIONAL SEMIGROUPS 31 Proof. By Lemma 4.9 we see that each Kj I and hence K I. The Corollary then follows from Property 4.1. Corollary 4.4 ([30], Corollary 2). We have Wj = Kj Wj . Proof. By Corollary 4.3 we get Kj Wj Wj . We also have Wj = Wj \ Wj Wj \ Wj = (Wj Wj ) \ Wj = (Wj Kj Wj ) \ Wj = Kj Wj . Lemma 4.10 ([30], Lemma 7). We have f (K) K for all f G. Proof. Let z Kj be such that Wj works for z. Suppose that Wl = f (Wj ) = W0 . So Wj contains no poles of f , else such a pole would be in Wj (by the complete invariance of W under f since W0 W and Lemma 4.9) and hence f (Wj ) = W0 . By the argument principle, f ( ) Wl winds around f (z), thus f (z) Kl as f (z) Wl by the complete invariance of W under the map f . Note / that f ( ) might not work for f (z) Kl since it might not be simple, but f (z) Kl since it cannot be in the unbounded component of C\Wl and have a curve in Wl , namely f ( ), wind around it. Now suppose that f (Wj ) = W0 . So ( f )(Wj ) = W0 is bounded and Wj contains no poles of f (else f (Wj ) = W1 ). So ( f )( ) winds around ( f )(z) and hence ( f )(z) K0 , i.e., f (z) K0 . So f (Kj ) K and hence we conclude f (K) K. Lemma 4.11 ([30], Lemma 8). We have for all f G, f (W ) W0 = . Also W N (G) and in particular K J(G) = . Proof. We have f (W ) = f (W K) = f (W ) f (K) W K = W . Since W W0 = (since W is open), Lemma 4.7 and Montel s Theorem nish the proof. Corollary 4.5 ([30], Corollary 3). We have J(G) Wj for each j = 0, . . . , L 1. Proof. This follows immediately from Lemma 4.8, Corollary 4.4 and Lemma 4.11. Remark 4.4. It is of interest to note that for any positive integer n there exist disjoint simply connected domains D1 , . . . , Dn in C with D1 = D2 = = Dn (see [18], p. 143). Thus Corollary 4.5 does not imply that L < 3 from a purely topological perspective. Lemma 4.12 ([30], Lemma 9). We have f 1 (K) K for all f G. Hence by Lemma 4.10, K is completely invariant under each f G. 32 DAVID BOYD AND RICH STANKEWITZ Proof. Let z Kj Wj and say f (w) = z. De ne Wk = f 1 (Wj ) by Remark 4.2. We obtain sequences zn Wj such that zn z, and wn Wk such that wn w and f (wn ) = zn . Hence we see that w Wk , else w Wk and z = f (w) Wj . If w Kk , then / w Wk by Corollary 4.4. Let be the component of Wj that contains f ( Wk ). Since z , the set must be one of the components of Kj . By Corollary 4.5 we see that there exists a Wk J(f ). Hence f ( ) Kj J(f ) which is a contradiction since we know by Lemma 4.11 that K is disjoint from J(G) J(f ). This contradiction implies w Kk and hence f 1 (K) K. Lemma 4.13 ([30], Lemma 10). If W has L components where 2 L < + , then each is simply connected. Proof. Since K and W are each completely invariant under each f G, so is W = W K. By Lemma 4.11 we see that C \ W is completely invariant under each f G, closed, and contains J(G). Hence I C \ W . This implies that W = W and hence each component of W is then simply connected. We are now able to present the proof of Theorem 4.1. Proof of Theorem 4.1. If W has L components where 2 L < + , then each is simply connected by Lemma 4.13. Select a map f G. Letting n 1 be selected so that each of the components Wj of W is completely invariant under f n , we get by the Riemann-Hurwitz relation (see [33], p. 7) f n (Wj ) = deg(f n ) 1 where we write g (B) = z B [vg (z) 1] and vg (z) is the valency of the map g at the point z. Hence we obtain L 1 L(deg(f ) 1) = n j=0 f n (Wj ) f n (C) = 2(deg(f n ) 1) and so L 2. The last equality follows from Theorem 2.7.1 in [5]. Remark 4.5. Note that if L = 2, then each component of W is necessarily simply connected. We know from iteration theory that each of the four possibilities (0, 1, 2, ) for the number of components of the set of normality can be achieved. So by constructing semigroups G such that all the elements have the same Julia set we know that the only four possibilities for the number of components of the completely invariant set of normality of DYNAMICS OF RATIONAL SEMIGROUPS 33 G can also be achieved. However, it does not seem possible that all four possibilities can be achieved if we restrict ourselves to the cases where two elements of the semigroup G have nonequal Julia sets. For example, if G contains two polynomials with nonequal Julia sets then the completely invariant set of normality is necessarily empty (see [32], Theorem 1). We do have the following examples however. 1 Example 4.3. Consider f (z) = 2z z . One can easily show that the extended real line R is completely invariant under f and that J(f ) is a Cantor subset of the interval [ 1, 1] (see [5], p. 21). Let (z) = i 1+z , 1 z 2 1 h(z) = z 2 , and set g(z) = ( h 1 )(z) = z 2z . Hence J(g) = (J(h)) = R, (see [5], p. 50). So we see that I( f, g ) = R = C, but J(f ) J(g). Note that J(f ) J(g) = I in this example. We also point out that J( f, g ) = R = I. 1 Example 4.4. Consider f (z) = 2z z as in Example 4.3. Let (z) = 1 z + 1, and set g(z) = ( f 1 )(z) = 2z z 1 1. Claim 4.1. In Example 4.4 we have J( f, g ) = [ 1, 2] and I(G) = R. Proof. De ne A = [ 1, 2]. Since f is a strictly increasing map of each of the intervals 1 1+ 3 1 3 A1 = 1, A and A2 = , A 2 2 2 onto the interval A, we can de ne two branches, say f1 and f2 , of f 1 on A by f1 (A) = A1 and f2 (A) = A2 . As |f (z)| > 2 on A1 and A2 , we see that f1 and f2 are contractions on A. Since g is a strictly increasing map of each of the intervals 1+ 3 1 3 1 , A and A4 = ,2 A A3 = 2 2 2 onto the interval A = [ 1, 2], we can de ne two branches, say g1 and g2 , of g 1 on A by g1 (A) = A3 and g2 (A) = A4 . As |g (z)| > 2 on A3 and A4 , we see that g1 and g2 are contractions on A. We note that A is backward invariant under both f and g since Aj A for 1 j 4, and so J( f, g ) A = [ 1, 2]. We next note that we can de ne an iterated function system on A using the functions f1 , f2 , g1 , and g2 . Let W (X) = f1 (X) f2 (X) g1 (X) g2 (X) for any compact subset X A. We note that W (A) = A and so by Iterated Function Systems (IFS) theory, A is the unique attractor set for this IFS. Let B = J(f ) J(g) and note that by the backward invariance of J( f, g ) we get W n (B) J( f, g ) for all n. 34 DAVID BOYD AND RICH STANKEWITZ Since J( f, g ) is closed and W n (B) A in the Hausdor metric, we see that A J( f, g ). Since [ 1, 2] = J(G) I(G) and R is completely invariant under both f and g, we see by Lemma 4.3 that I(G) = R. So we see that it is possible for a completely invariant set of normality of a semigroup G which contains two elements with nonequal Julia sets, to have 0 or exactly 2 components. We feel that the interplay between functions with nonequal Julia sets and the fact that if I(G) has interior then I(G) = C demands that only under special circumstances can we have W (G) be nonempty, when two elements of the semigroup G have nonequal Julia sets. We state the following conjectures which are due to Aimo Hinkkanen and Gaven Martin. Conjecture 4.2. If G is a rational semigroup which contains two maps f and g such that J(f ) = J(g) and I(G) = C, then W (G) has exactly two components, each of which is simply connected, and I(G) is equal to the boundary of each of these components. Conjecture 4.3. If G is a rational semigroup which contains two maps f and g such that J(f ) = J(g) and I(G) = C, then I(G) is a simple closed curve in C. Of course Conjecture 4.2 would follow from Conjecture 4.3. We nish by including some comments on the number of components of the set of normality N (G) of a rational semigroup G. It is not known if the set N (G) must have only 0, 1, 2, or in nitely many components when G is a nitely generated rational semigroup. However, for each positive integer n, an example of an in nitely generated polynomial semigroup G can be constructed with the property that N (G) has exactly n components. These examples were constructed by David Boyd in [6]. 4.2. Polynomial semigroups. The material from this section comes entirely from [32]. When the semigroup G contains only elements with the same Julia set J, then we have seen that I(G) = J = J(G). If, however, there are two functions with nonequal Julia sets, then we do not expect that J(G) should necessarily equal I(G), see Example 1.1. For example, if the functions with nonequal Julia sets are polynomials, then we will show that I(G) must coincide with the entire Riemann sphere. Speci cally, we prove the following theorems. Theorem 4.2 ([32], Theorem 1). For polynomials f and g of degree greater than or equal to two, J(f ) = J(g) implies I(G) = C where G = f, g . DYNAMICS OF RATIONAL SEMIGROUPS 35 The following theorem follows immediately. Theorem 4.3 ([32], Theorem 2). For a rational semigroup G which contains two polynomials f and g of degree greater than or equal to two, J(f ) = J(g) implies I(G ) = C. We rst establish the necessary lemmas. Lemma 4.14 ([32], Lemma 4). If f and g are polynomials of degree greater than or equal to two and J(f ) = J(g), then I. Proof. Denoting the unbounded components of the respective Fatou sets of f and g by F and G , we recall (see [5], p. 54 and p. 82) that J(f ) = F and J(g) = G . Since F and G are domains with nonempty intersection and F = G , we have J(f ) G = or J(g) F = . Hence we may select z J(g) F , say. Denoting the nth iterate of f by f n , we see that f n (z) , and by the forward invariance under the map f of the set I we get that each f n (z) I. Since I is closed we see then that I. Remark 4.6. Since it will be necessary later, we make special note of the fact used in the above proof that J(f ) = J(g) implies J(f ) G = or J(g) F = . Remark 4.7. Note that the proof above shows also that is not an isolated point of I when J(f ) = J(g). This, of course, also follows from Corollary 4.1 and Lemma 4.14. The disc centered at the point z with radius r will be denoted (z, r). Lemma 4.15 ([32], Lemma 5). Suppose that (0, r ) = A B where A is open, A and B are disjoint, and both A and B are nonempty. If both A and B are completely invariant under the map L(z) = z j de ned on (0, r ) where 0 < r < 1 and j 2, then the set A is a union of open annuli centered at the origin and hence B is a union of circles centered at the origin. Furthermore, each of A and B contains a sequence of circles tending to zero. Proof. Let z0 = rei A. Since A is open we may choose > 0 such that the arc z0 = {rei : | | 2 } A. n Fix a positive integer n such that j n > 2 . Since Ln (z) = z j we get n Ln ( z0 ) = C(0, r j ) where C(z, r) = { : | z| = r}. n By the forward of invariance A under L, we see that C(0, r j ) A. But now by the backward invariance of A, we get C(0, r) = L n (C(0, r j )) A. n 36 DAVID BOYD AND RICH STANKEWITZ Thus for any rei A, we have C(0, r) A. Hence A, being open, must be a union of open annuli centered at the origin and therefore B, being the complement of A in (0, r ), must be a union of circles centered at the origin. n We also note that if C(0, r) A, then C(0, r j ) A is a sequence of circles tending to zero. Similarly we obtain a sequence of circles in B tending to zero. Lemma 4.16 ([32], Lemma 6). Let L : (0, r ) (0, r ), where 0 < r < 1, be an analytic function such that L(0) = 0. Let B be a set with empty interior which is a union of circles centered at the origin and which contains a sequence of circles tending to zero. If B is forward invariant under the map L, then L is of the form L(z) = az j for some non-zero complex number a and some positive integer j. Proof. Since L(0) = 0, we have, near z = 0, L(z) = az j + a1 z j+1 + a1 = az j (1 + z + ) a for some non-zero complex number a and some positive integer j. Let h(z) = L(z)/az j and note that h(z) is analytic and tends to 1 as z tends to 0. We shall prove that h(z) 1 and the lemma then follows. Let Cn = C(0, rn ) be sequence of circles contained in B with rn 0. We claim that each L(Cn ) is contained in another circle centered at the origin of, say, radius rn . If not, then the connected set L(Cn ) would contain points of all moduli between, say, r and r . This, however, would imply that B would contain the annulus between the circles C(0, r ) and C(0, r ). Thus we have L(Cn ) C(0, rn ). j So we see then that h(Cn ) C(0, rn /|a|rn ). But for large n we see that if h were non-constant, then h(Cn ) would be a path which stays near h(0) = 1 and winds around h(0) = 1. Since h(Cn ) is contained in a circle centered at the origin, this cannot happen. We thus conclude that h is constant. Lemma 4.17 ([32], Lemma 7). If B (0, r ) for 0 < r < 1 is a nonempty relatively closed set which is completely invariant under the maps H : z z j and K : z az m de ned on (0, r ) where a is a nonzero complex number and j, m are integers with j, m 2, then B = (0, r ) or |a| = 1. Proof. We may assume that |a| 1 by the following reasoning. Suppose that |a| 1. Let b be a complex number such that bm 1 = a and DYNAMICS OF RATIONAL SEMIGROUPS 37 de ne (z) = bz. Since H 1 (z) = z j /bj 1 and K 1 (z) = z m , we see that the lemma would then imply that (B) = (0, |b|r ) or |b| = 1. Since we know that (B) = (0, |b|r ) exactly when B = (0, r ), and |b| = 1 exactly when |a| = 1, we may then assume that |a| 1. We will assume that |a| < 1 and show that this then implies that B = (0, r ). We rst note that by Lemma 4.15, B is a union of circles centered at the origin and B contains a sequence of circles tending to zero. If C(0, ) B, then by the forward invariance of B under H, we see that C(0, j ) B. Also we get that if C(0, ) B, then by the forward invariance of B under K, we have C(0, |a| m ) B. Using a change of coordinates r = log this invariance can be stated in terms of the new functions (4.8) t(r) = jr and s(r) = mr + c where c = log |a| < 0. So the action of H and K on (0, r ) is replaced by the action of t and s on I = [ , log r ), respectively. In particular, we de ne B = {log : C(0, ) B} { } keeping in mind that B is a union of circles centered at the origin. Then (4.9) (4.10) (4.11) (4.12) (4.13) s(B ) B , t(B ) B , s 1 (B ) I B , t 1 (B ) I B , B is closed in the relative topology on I. In order to make calculations a bit easier we rewrite s(r) = r0 + m(r r0 ) where r0 = c/(m 1) > 0. Hence s n (r) = r0 + m n (r r0 ), sn (r) = r0 + mn (r r0 ), tn (r) = j n r, t n (r) = j n r. 38 DAVID BOYD AND RICH STANKEWITZ Consider (t n s n tn sn )(r) = r r0 + Let dn = r0 r0 r0 + n n n. jn m mj r0 r0 mn + j n 1 r0 + n n n = r0 jn m mj mn j n and note that 0 < dn r0 with dn 0 as n . We also note that (t n s n tn sn )(r) = r r0 + dn implies that n (s t n sn tn )(r) = r +r0 dn since these two functions are inverses of each other. We claim that ( , log r r0 ] B . Let us suppose that this is not the case, and suppose that (r , r) is an interval disjoint from B with < r < r log r r0 . Since B is a closed subset of [ , log r r0 ], we may assume that this interval is expanded so that r B . Note that here we used the fact that B contains a sequence of circles going to 0, hence B contains a sequence of points going to . Let rn = (t n s n tn sn )(r ) = r r0 + dn . We claim that each rn is in B . This is almost obvious from the invariance of B under s and t in (4.9) through (4.12), but some care needs to be taken to insure that each application of s, t, s 1 , and t 1 takes points to the right domain. By (4.9) we see that s(r ), s2 (r ), . . . , sn (r ) B . Hence by (4.11) we get (t sn )(r ), (t2 sn )(r ), . . . , (tn sn )(r ) B . Since s 1 (r) > r for r ( , r0 ) we see that because (s n tn sn )(r ) is clearly less than r (as t(r) < r for r ( , 0)), also each of (s 1 tn sn )(r ), . . . , (s n tn sn )(r ) must be less than r < log r . Hence by (4.10) we see that each of these points lies in B . Similarly, since t 1 (r) > r for r ( , 0) and (t n s n tn n s )(r ) = r r0 + dn r < log r < 0, also each of (t 1 s n tn sn )(r ), . . . , (t n s n tn sn )(r ) lies in I = [ , log r ). Hence by (4.12) each of these points is in B and so each rn B . Hence we conclude that r r0 B since B is relatively closed in I and rn r r0 I. Note also that rn r r0 . Now we claim that for any r B ( , log r r0 ), we have r + r0 B . Let rn = (s n t n sn tn )(r ) = r + r0 dn . Noting that each rn < r + r0 < log r we may again use the invariance of B under s and t in (4.9) through (4.12) in a similar fashion as above to obtain that each rn B . Thus also the limit r + r0 B . Consider again rn r r0 . By applying the above claim to each rn r < log r r0 , we get that each rn + r0 B . Since rn + r0 r we then see that we have contradicted the statement that (r , r) is disjoint from B . DYNAMICS OF RATIONAL SEMIGROUPS 39 So we conclude that ( , log r r0 ] B . Clearly then by the partial backward invariance of B under the map t we get [ , log r ) B . Hence we conclude that (0, r ) = B. In order to avoid some technical di culties we will make use of the following well known result. Theorem 4.1. A polynomial f of degree k is conjugate near to the map z z k near the origin. More speci cally, there exists a neighborhood U of such that we have a univalent :U Proof. After conjugating f by z 1/z we may apply Theorem 6.10.1 in [5], p. 150 to obtain the desired result. We will denote the conjugate function of f by F , i.e., In order to further simplify some of the following proofs we will assume that (U ) = D = (0, r ). Note that U is forward invariant under f since D = (0, r ) is forward invariant under F . We may and will also assume that U is forward invariant under g as well. We now de ne a corresponding function for g using the same conjugating map as we did for f . Let G be the function de ned on D = (0, r ) given by G = g 1 . Note that G(D) D. Via this change of coordinates, we will use the mappings F and G to obtain information about the mappings f and g. In transferring to this simpler coordinate system we make the following de nitions. Let W denote the image of W under , i.e., W = (U W ). Let I denote the image of I under , i.e., I = (U I). Thus W is open and I is closed in the relative topology of D. Note that W and I are disjoint since W and I are disjoint and is univalent. Also since W I = C it easily follows that W I = (U ) = D. By the forward invariance of W U under f we see that (4.14) F (W ) = F (W U ) = f (W U ) (W U ) = W . Similarly we get F (I ) I . F (z) = f 1 (z) = z k . (0, r ) for 0 < r < 1 with ( ) = 0 and f 1 (z) = z k . (4.15) Since I and W are disjoint and forward invariant under F , and since I W = D, we see that 40 DAVID BOYD AND RICH STANKEWITZ (4.16) (4.17) Note that in the same way as we obtained the results for F we get (4.18) (4.19) (4.20) (4.21) G(W ) W , G 1 (I ) D I , G(I ) I , F 1 (W ) D W . F 1 (I ) D I , Lemma 4.18 ([32], Lemma 8). If G(z) = az l with |a| = 1, then J(f ) = J(g). Proof. The proof relies on the use of Green s functions. It is well known that the unbounded components F and G support Green s functions with pole at which we will denote by Gf and Gg respectively. It is also well known that on U we have since is a map which conjugates f to z z k (see [5], p. 206). Since for (z) = bz where bl 1 = a, the function conjugates g in U to z z l , we get in U , where the last equality uses the fact that |a| = 1, and so |b| = 1. Hence Gf = Gg in U . Since Gf and Gg are each harmonic away from we get that Gf = Gg on the unbounded component C of F G . We claim that this implies that J(f ) = J(g). Assuming that J(f ) = J(g), we see by Remark 4.6 that there exists a point which lies in the Julia set of one function, yet in the unbounded component of the Fatou set of the other function. Let us therefore suppose that z0 J(g) F . Let be a path in F connecting z0 to . We see that must intersect C somewhere, say at z0 . Since z0 F C we get z0 G = J(g). We may select a sequence zn C such that zn z0 . Since z0 lies on the boundary of the domain of the Green s function Gg , i.e., z0 J(g), we have Gg (zn ) 0 (see [5], p.207). Since z0 lies in the domain of the Green s function Gf we see that Gf (zn ) Gf (z0 ) > 0. We cannot have both of these happen since Gf (zn ) = Gg (zn ) and so we conclude that J(g) F = . Hence we conclude that J(f ) = J(g). We now are able to prove Theorem 4.2. Gg (z) = log | (z)| = log |b (z)| = log | (z)| Gf (z) = log | (z)| G 1 (W ) D W . DYNAMICS OF RATIONAL SEMIGROUPS 41 Proof of Theorem 4.2. Consider whether or not I has nonempty interior. If I o = , then by Lemma 4.1 we get I = C. If I o = , then Lemma 4.15 implies that the set W is a union of open annuli centered at the origin and hence I is a union of circles centered at the origin. Since I o = , the set I has empty interior. Since we know by Remark 4.7 that there exists a sequence of points in I tending to in nity when J(f ) = J(g), also I must contain a corresponding sequence of circles tending to zero. By Lemma 4.16 we see that the function G is of the form G(z) = az l for some non-zero complex number a. By considering the set I , we see that Lemma 4.17 implies that |a| = 1. We see that Lemma 4.18 then implies J(f ) = J(g). 5. An Invariant measure for finitely generated rational semigroups As stated in Proposition 1.4, the Julia set of a rational semigroup is contained in the set of accumulation points of the backward orbit of any non-exceptional point a. When a J(G), we have in fact that J(G) = O (a). When G is nitely generated, this serves as the basis for a computer algorithm for making an approximate picture of the Julia set. See [26], pp. 35 38 and [24], Appendix E for a discussion of the single generator case. Many are familiar with these pictures in the classical cyclic semigroup case. Experimental evidence indicates that while this procedure often does yield a believable picture, there are certain phenomena which prevent this nite process from giving a complete picture. Consider the following construction. Let f be a rational function of with deg f = d 2 and let a C \ E(f ). Then for n 1 de ne n = a 1 dn z f n (z)=a where z is the unit point mass measure at z and the sum is taken over all solutions to f n (z) = a counted according to multiplicity. As there are exactly dn such solutions, the measure n is a probability a measure. Thus a is the probability measure evenly distributed (up to n multiplicity) over the preimages of a under f n . The following result was established by Lyubich in [20]. 42 DAVID BOYD AND RICH STANKEWITZ a Theorem 5.1 (Lyubich). The measures n converge weakly to a unique regular Borel probability measure = f independently and locally uniformly in a C \ E(f ). The closed support of is J(f ). Further, satis es the following properties. For any Borel set E, we have (E) = f 1 (E) . 1 (E) (f (E)) d where equality holds if f is injective on E. Recall that a sequence of measures n on a space X converges weakly to the measure if d n d for every continuous function on X. Roughly, the measure has the largest concentration of its support on the part of the Julia set that is best approximated by the above mentioned computer scheme. There is much known about this measure . We list a few facts here. For a polynomial f , Brolin showed in [9] that the measure is the harmonic measure of J(f ) as seen from in nity. Lyubich showed that is the measure of maximal entropy for the function f . Results similar to those of Lyubich were established independently by A. Freire, A. Lopes, and R. Ma in [19] and [21]. ne For a nitely generated rational semigroup G, a similar computer scheme can be implemented to create an approximate picture of J(G). As in the cyclic case there are observable instances where no reasonable number of iterations in the computer algorithm will ll in large areas known to belong to the Julia set. For example, consider G = f, g where f (z) = z 2 + 2z and g(z) = z 2 + z/2. It is easy to check that 1 is a superattracting xed point for f and that g( 1) = 1/2 is a repelling xed point for g. Hence the full component of N (f ) containing 1, which in this case is the disk of radius 1 centered at 1, is contained in J(G). See Corollary 2.1. However Figure 1 indicates a large gap in the picture near 1, and the same gap appears in very high numbers of iterations in the program. A natural question is to what extent does Lyubich s result generalize to rational semigroups? It turns out that the results substantially go through but with some important di erences. The discussion below is in uenced by Steinmetz s presentation of Lyubich s result in [33]. Let G = f1 , . . . , fk be a nitely generated rational semigroup with deg fj = dj 2. We remark that in this setting the semigroup has a best generating set. We say that a generating set is minimal if no We further have DYNAMICS OF RATIONAL SEMIGROUPS 43 1.5 1 0.5 -2 -1.5 -1 -0.5 0.5 -0.5 -1 -1.5 Figure 1. Julia set of z 2 + 2z, z 2 + z/2 generator can be expressed as a word in the remaining generators. We have the following result, whose proof is left as an exercise. Lemma 5.1 ([7], Lemma 2). Every nitely generated rational semigroup G such that deg g 2 for all g G has a unique minimal generating set. While the statements of the following results hold for an arbitrary generating set, the conditions of some are likely only to be satis ed by the minimal generating set, and hence from now on, when we refer to the generating set for a nitely generated rational semigroup, we will assume that it is the minimal generating set. Let a C \ E(G). For any integer n 1 we de ne (5.1) a = n 1 dn z g(z)=a l(g)=n where here d = d1 + + dk and the sum is taken over all solutions, counted according to multiplicity, to the equations g(z) = a as g ranges 44 DAVID BOYD AND RICH STANKEWITZ over all words in G of length n. Since there are exactly dn such solutions, the measure a is a probability measure. (Note that in this n sum we may have multiple words representing the same group element. This is ne, as this mirrors how the computer algorithm would work. In most cases this will not be an issue as an arbitrary rational semigroup is likely to be free on its generating set.) The results we wish to discuss follow. Theorem 5.2 ([7], Theorem 1). Let G = f1 , . . . , fk be a nitely generated rational semigroup with deg fj = dj 2 and d = d1 + + dk . Then the measures a de ned by (5.1) converge weakly to a regular n Borel probability measure = G independently of and locally uniformly in a C \ E(G). The closed support of is J(G). Further, satis es the following inequalities. For any Borel set E C, k (5.2) (E) + i=1 1 fi d k fj 1 (E) j=1 k \E fj 1 (E) j=1 k (E) + and also (5.3) (E) 1 d k i=1 dj fi d k fj 1 (E) j=1 \E (fj (E)) . j=1 We also have the following corollary, indicating conditions that guarantee that the measure G is invariant under the generating set of G. Corollary 5.1 ([7], Corollary 1). The equalities k k (5.4) (E) = j=1 fj 1 (E) = j=1 (fj 1 (E)) hold for every Borel set E J(G) if for all integers 1 i, j k, i = j, fi 1 (J(G)) fj 1 (J(G)) = 0. 5.1. Discussion of the inequalities (5.2). Given a Borel set E C, a point a C \ E(G), and a positive integer n, the measure a (E) is n the proportion of the total number of preimages of a under length n k 1 a words of G that lie in E. Consider the measure n+1 j=1 fj (E) . DYNAMICS OF RATIONAL SEMIGROUPS 45 Each preimage of a under a length n word lying in E has a total of d preimages under the generators fj that lie in k fj 1 (E). Thus from j=1 (5.1) we see that k (5.5) a (E) n a n+1 j=1 fj 1 (E) However, it is possible that some preimage of a under a length n word that lies outside of E will itself have a preimage in k fj 1 (E) under j=1 some generator, assuming that k, the number of generators, is at least 2. Hence the inequality in (5.5) may be strict. The sums found in (5.2) represent a lower and upper bound on this error for the limiting measure. 5.2. Discussion and Proof of Corollary 5.1. We rst note that as it is possible that fj 1 (N (G)) J(G) = for some generator fj , we cannot expect (5.4) to hold for every Borel set E, since (N (G)) = 0 and the -measure of any open set meeting J(G) is positive. Thus some restriction is necessary. We give a proof of Corollary 5.1, assuming the truth of Theorem 5.2. k 1 Examining the inequalities (5.2), we see that (E) = j=1 fj (E) if and only if fi that k j=1 fj 1 (E) \ E = 0 for i = 1, . . . , k. Assume for all i = j. Hence given any Borel subset E J(G), and l {1, . . . , k}, it follows that we need to show k fi 1 (J(G)) fj 1 (J(G)) = 0 (5.6) fl j=1 fj 1 (E) \E = 0. It su ces to show that k fl j=1 fj 1 (E) \E J(G) has -measure 0. However, k fl j=1 fj 1 (E) \ E J(G) k j=1 j=l fl (fj 1 (E)) J(G) since any point in fl for j = l. Any point in fl (fj 1 (E)) J(G) is the image under fl of a fj 1 (E) \ E must lie in some fl (fj 1 (E)) 46 DAVID BOYD AND RICH STANKEWITZ point in fj 1 (E) fl 1 (J(G)). Since E J(G), we have shown that j=l fl (fj 1 (E)) J(G) j=l fl fj 1 (J(G)) fl 1 (J(G)) . We are assuming that fj 1 (J(G)) fl 1 (J(G)) has -measure 0 for all j = l. By examining the inequality (5.3) of Theorem 5.2 we may conclude that if (F ) = 0 for any Borel set F , then (g(F )) = 0 for all g G. Hence the set j=l fl fj 1 (J(G)) fl 1 (J(G)) also has -measure 0. The above inclusions now imply (5.6) and so we k 1 also have (E) = j=1 fj (E) as claimed. It is an easy exercise to show that k k j=1 fj 1 (E) = j=1 (fj 1 (E)) under the assumptions of the theorem. Remark 5.1. We believe that the su cient conditions of the corollary are also necessary, but as of this writing a complete proof has not been established. Example 5.1. Let f (z) = z 2 and let g(z) = z 2 /a for some a > 1. Let G = f, g . It is shown in [15], Example 1, that J(G) = {z : 1 |z| a}. We explicitly construct for this semigroup and show that satis es the conditions of Corollary 5.1. log( a)+i The preimages of z0 = a = e under f and g are We inductively calculate the preimages of z0 under length n words. Assume that the preimages of z0 under the length n words of G are 2j 1 (2k 1) n log a + i zj,k = exp n 2 2n n for j, k = 1, . . . , 2n . The preimages under f and g of a given point zj,k are 2j 1 (2k 1) exp log a + i , n+1 2 2n+1 (2k 1) 2j 1 log a + i + i , exp 2n+1 2n+1 2j 1 (2k 1) log a + log a + i , exp n+1 2 2n+1 1 1 3 3 3 3 {e 2 log( a)+i 2 , e 2 log( a)+i 2 , e 2 log( a)+i 2 , e 2 log( a)+i 2 }. DYNAMICS OF RATIONAL SEMIGROUPS 47 exp 2j 1 (2k 1) log a + log a i + i 2n+1 2n+1 . After reordering, induction yields that the preimages of z0 under the length n words of G for all n 1 are n zj,k = exp (2k 1) 2j 1 log a + i n 2 2n 2n n zj,k . for j, k = 1, . . . , 2n . Thus a n 1 =n 4 j,k=1 Let We think of R as the set log(J(G)). Let n n wj,k = log(zj,k ) = R = {w = u + iv : 0 u log a, 0 v < 2 }. (2k 1) 2j 1 log(a) + i n+1 2 2n 2n n wj,k . and let 1 n = n 4 j,k=1 Note that for any set E J(G), we have that n a (E) = n (log(E)). The measures n converge weakly to m/(2 log(a)) where m is Lebesgue measure restricted to R. To see this one need just consider the de nition of the Riemann integral. This implies that (E) = m(log(E)) 2 log(a) where is the measure from the conclusion of Theorem 5.2. Since f 1 (J(G)) g 1 (J(G)) = {z : |z| = a}, we have (f 1 (J(G)) g 1 (J(G))) = 0 and so G satis es the conditions of Corollary 5.1. We further remark that it is easy to construct examples of rational semigroups G = f1 , . . . , fk where fi 1 (J(G)) fj 1 (J(G)) = for all i = j. Such G clearly satisfy the conditions of Corollary 5.1. 5.3. Proof of Theorem 5.2. We break the proof up into several parts, dealing with each of the statements of Theorem 5.2 separately. 48 DAVID BOYD AND RICH STANKEWITZ 5.4. Proof of the weak convergence of a . Consider the Banach n space C(K) of continuous real valued functions on a compact set K C \ E(G) with norm = max{| (z)| : z K}. For our purposes it will su ce to consider such K where K contains at least three points and where fj 1 (K) K for each generator fj . These assumptions guarantee that g 1 (K) K for all g G and hence J(G) K by Remark 1.1. For z K, de ne the function (5.7) (T )(z) = K ( )d z ( ) 1 1 = d d (zj ) j=1 where the points zj are the solutions of fi (w) = z, listed according to multiplicity, for all i with i = 1, . . . , k. The function T is continuous on the compact set K. For if > 0 is given, we may choose 1 > 0 so that if q(a, b) < 1 , where q( , ) is the chordal metric, and a, b K then | (a) (b)| < and we may further choose 2 > 0 so that if q(z, z ) < 2 the solutions of the equations f (w) = z, f (w) = z may be ordered so that q(zj , zj ) < 1 . Then d |(T )(z) (T )(z )| d 1 j=1 | (zj ) (zj )| < . The action of T on C(K) is clearly linear, hence T : C(K) C(K) is a linear operator. By considering 1, it is immediate that the operator norm T of T satis es T = sup{ T : = 1} = 1. Hence T is a continuous linear operator from C(K) to itself. Recursively de ne T m via T m = T (T m 1 ). Then (5.8) 1 (T m )(z) = m d dm m (zj ) = j=1 K ( ) d z ( ) m m where here the points zj are the solutions to the equations g(w) = z, listed according to multiplicity where g ranges over the length m words of G. We see this as follows. Assume 5.8 holds for m 1. Then (T )(z) = (T (T m m 1 1 ))(z) = d d (T i=1 m 1 )(zi1 ) 1 = d d 1 dm 1 dm 1 m 1 (zi,k ) k=1 i=1 where the points zi1 are the solutions to f (w) = z under length one m 1 words of G and the points zi,k represent the solutions of the equations g(w) = zi1 where g ranges over the length m 1 words of G, which thus in total also represent the solutions of h(w) = z as h ranges DYNAMICS OF RATIONAL SEMIGROUPS 49 dm j=1 m (zj ) = over the length m words of G. Hence (T m )(z) = d m ( ) d z ( ) by induction. m K 5.5. Relationships between and T m . We establish some relationships between and T m when = 0. Recall that K is compact and backwards invariant under the generators of the semigroup. For a given integer m 1, choose z0 = z0 (m) K so that T m = |(T m )(z0 )|. Then 1 T = |(T )(z0 )| m d m m dm j=1 | ((z0 )j )| with equality if and only if ((z0 )j ) = where the sign depends only on m and not on j. We remark that if is not identically or on J(G), then there is an integer m such that T l < for all l m. We see this as follows. If there is a point w J(G) and a neighborhood U of w such that | (u)| < for all u U K, then, since w J(G), Proposition 1.6 implies that there exists an integer N such that for all n N , we have K l(g)=n g(U ). In particular, if m N there is a solution (z0 )j to g(w) = z0 in U for some word g of length m. Similarly if there are points w1 , w2 J(G) with (w1 ) = and (w2 ) = , then there are disjoint neighborhoods Ui of wi on which | | is close to respectively, such that for each i the equation g(z) = z0 has a solution in Ui for some word g of length m. Hence in either case we see that if is not identically or on J(G), then T m < for this and all larger m. Clearly for constant , we have T m = for all m. Similarly we see that the minimal value of T m is nondecreasing in m as follows. Choose z0 K so that min{(T )(z) : z K} = (T )(z0 ). Then min{(T )(z) : z K} = (T )(z0 ) = 1 d d j=1 ((z0 )j ) min{ (z) : z K}. Further, equality holds if and only if ((z0 )j ) = min{ (z) : z K} for all j. Then by induction, for all m 1, we have that min{(T m+1 )(z) : z K} min{(T m )(z) : z K} min{ (z) : z K}. Our goal is to show that the sequence T m converges uniformly on K to a constant function l . To do this we need the following lemma. Lemma 5.2. For every C(K), the family {T m : m = 1, 2, . . . } is equicontinuous. 50 DAVID BOYD AND RICH STANKEWITZ The proof of Lemma 5.2, which is somewhat technical, comes later. For now assume that the lemma has been established. 5.6. Convergence of T m . Given the lemma, we show the uniform convergence of the functions T m as follows. The Arzel` Ascoli Thea m orem gives that every subsequence of T has a uniformly convergent subsequence which must converge to a continuous function. Thus assuming that T nk converges uniformly on K to the continuous function and by passing to a further subsequence, if necessary, we may assume that mk = nk+1 nk and that T mk converges uniformly to a continuous function . Then Thus = , i.e., T mk . It follows from previous considerations that for all m, T m . Also for mk m, = lim mk T nk+1 T mk = T mk (T nk ) T nk 0. T mk = lim mk T mk m (T m ) T m hence T m = for all m. We may then conclude, as noted above, that either or on J(G). For convenience, assume that on J(G). We will show that in fact, on all of K. Assume that this is not the case. (We are necessarily now assuming that K \ J(G) = .) Then Recall that T does not decrease the minimum value. Hence we have that min{(T mk )(z) : z K} is nondecreasing in k. This implies that since T mk , we must have that min{(T mk )(z) : z K} = c for all k. Choose zk K so that T mk (zk ) = c. First assume that in nitely many zk are the same point. By passing to a subsequence if necessary, we may then assume that zk = z0 for all k. As we have seen before, (T mk )(z0 ) = min{ (z) : z K} = c if and only if (z) = c on every preimage of z0 under all words of G of length mk . Recall that we are assuming on J(G). Let U be a neighborhood of J(G) so that (z) > c for all z U K. By Proposition 1.6 there exists an integer N such that z0 l(g)=n g(U ) for all integers n N . In particular, z0 has a preimage in U under some length mk word when mk N . Hence (T mk )(z0 ) > c which is a contradiction. For the next case, since K is compact, by passing to a subsequence if necessary, we may assume that zk z0 K. Take a small closed neighborhood D c = min{ (z) : z K} < . DYNAMICS OF RATIONAL SEMIGROUPS 51 of z0 in K. Again by Proposition 1.6 there exists an integer N such that D g(U ) l(g)=n for n N . Assume that zk D for all mk N1 N . Then zk has a preimage in U under some length mk word whenever mk N1 . Again this contradicts the assumption that T mk (zk ) = c. Thus l is constant. Further, for m > nk , so we can see that T m l uniformly on K as m , as claimed. 5.7. The existence and regularity of . For functions and continuous on K, we have |l l | = lim m T m l = T m nk (T nk ( l )) T nk ( l ) 0 T m ( ) , l may be considered as a continuous linear functional on C(K) and by the Riesz Representation Theorem may be represented uniquely in the form l = K ( )d K ( ) for a regular Borel measure K . Hence for all C(K) and for all a K, m lim K ( ) d a ( ) = m ( ) d K ( ) K where the convergence is uniform in a K. However, as J(G) K for all sets K under consideration, we see that K = is independent of K and so can be considered as a measure on C. Setting 1, we see that is a probability measure. This will complete our proof that the measures a converge weakly to the regular Borel probability measure n , independently of and localy uniformly in a, once we have proven Lemma 5.2. 5.8. Proof of Lemma 5.2. We now proceed with a proof of Lemma 5.2. It su ces to prove local equicontinuity. Let Cj denote the set of critical points of fj . Then CV1 = k fj (Cj ) j=1 is the set of the critical values of the length one words of G, k k k CV2 = j1 ,j2 =1 fj2 (fj1 (Cj1 )) fj (Cj ) = j=1 j=1 fj (CV1 ) CV1 52 DAVID BOYD AND RICH STANKEWITZ is the set of the critical values of the length two words of G and in general, k n k CVn = j=1 fj (CVn 1 ) CVn 1 = m=1 j1 ,j2 ,...,jm =1 (fjm fj1 )(Cj1 ) is the set of the critical values of the length n words of G. Let U be a simply connected domain. Assume that CVl is disjoint from U . Then for n l there are exactly dn single valued analytic inverse functions on U from length n elements of G. In general, let n (U ) denote the total number of single valued analytic inverse functions on U from the length n words of G. Let denote the total number of distinct critical values of the functions fj . We claim that for all n, n l (5.9) d n k n l j=1 dj k n l j . The claim is clearly true for n l. For a given function fi1 fin , let (i1 ,...,in ) (U ) denote the number of its single valued analytic inverse functions on U . Then n (U ) = k1 ,...,in =1 (i1 ,...,in ) (U ). The images of i U under the inverses of fi1 fin are mutually disjoint and simply connected. Then at least (i1 ,...,in ) of them contain no critical value of any fi . Thus fi1 fin fi has (i1 ,...,in ,i) (U ) di ( (i1 ,...,in ) (U ) ) inverses on U . In particular, k k k k n+1 = i=1 i1 ,...,in (i1 ,...,in ,i) i=1 i1 ,...,in di ( (i1 ,...,in ) ) = d( n k n ). n l Then by induction, n+1 d( n k n ) d(dn k l n+1 l j=1 dj k n l j k n ) =d n+1 k l j=1 dj k n+1 l j which gives the claim. Note that this estimate depends on U only in the fact that U misses CVl . Let > 0 be given. There is a positive integer L depending on so that if U is a simply connected domain that misses CVL , then for all n 1, dn n (U ) dn k d L n L j=1 k d n L j k d L i=0 k d i DYNAMICS OF RATIONAL SEMIGROUPS 53 = k d L recalling that d 2k. Case 1. Assume that z0 K and z0 CVL . Let U = (z0 , 2 ) be / a chordal neighborhood of z0 that misses CVL . Let hn , . . . , hnn denote 1 the single valued analytic inverses on U from the length n elements of G. It is shown in [17], Corollary 2.2, that the collection {hn : n j 1, 1 j n } forms a normal family on U . Hence given > 0, we may choose > 0 so that if z, K, q(z, ) < , q(z, z0 ) < , and n q( , z0 ) < , then q(hn (z), hj ( )) < for all j and n. Further assume j that was chosen so that | (u) (v)| < /2 whenever q(u, v) < and u, v K. Thus when z, K, q(z, ) < , q(z, z0 ) < , and q( , z0 ) < , we have n d d < , d k 4 |(T )(z) (T )( )| = |d n n n n j=1 ( (zj ) ( j ))| dn n <. dn d n j=1 n | (hn (z)) (hj ( ))| + 2 j We make the remark that this technique will work for < if one looks at points in K \ CVM for a su ciently large choice of M > L. Case 2. Now assume that z0 K CVL . Choose M > L so that for all n 1, as in Case 1, for points u, v in suitable neighborhoods of points in K \ CVM . Recall that CVL CVM whenever L < M . As K CVM is nite, one can show that there exists an integer such that for any a K, there exists a word g of length such that g(z) = a has at least one solution outside of K CVM . Recall that g 1 (K) K for all g G. We wish to nd, as in Case 1, a > 0 such that if u, v K, q(u, z0 ) < /2, and q(v, z0 ) < /2, then for all n 1. Given a positive integer n, let Sn denote the set of solutions, listed according to multiplicity, of g(z) = z0 where g ranges over the length n words of G. The cardinality of Sn is dn . We inductively divide the sets Sn as follows. The integer was chosen such that there is at least one solution in S1 outside of K CVM . Let S1 (1) consist of this single point and |(T n )(u) (T n )(v)| < |(T n )(u) (T n )(v)| < /2 54 DAVID BOYD AND RICH STANKEWITZ let S1 (2) consist of the remaining d 1 points. For n = 2, let S2 (1) be the d preimages of S1 (1), let S2 (2) be the d 1 preimages under length words of the points in S1 (2) guaranteed to be outside of K CVM , and let S2 (3) be the remaining (d 1)2 points in S2 . Note that d + (d 1) + (d 1)2 = d2 . Suppose that the sets Sl have been broken into sets Sl (j) for 1 l n 1 and 1 j l + 1 where Sl (j) contains d(l j) (d 1)j 1 points for j = 1, . . . , l and Sl (l + 1) contains (d 1)l points. In particular, we assume that for j = 1, . . . , l 1, Sl (j) consists of the preimages of the set Sj (j) under words of length (l j) . We also assume that Sl (l) consists of the preimages of Sl 1 (l) under length words guaranteed to be outside of K CVM . We then de ne Sn (j) for 1 j n + 1 as follows. For j = 1, . . . , n 1, the set Sn (j) consists of the d(n j) (d 1)j 1 points which are preimages of the set Sn 1 (j) under length words. Note that Sn (j) consists of the preimage of Sj (j) under all length (n j) words. We de ne the set Sn (n) to consist of (d 1)n 1 points which are preimages of Sn 1 (n) guaranteed to be outside of K CVM . The set Sn (n + 1) then consists of the remaining (d 1)n preimages of Sn 1 (n). Note that the sum of the number of elements in the sets Sn (j) is given by n j=1 d(n j) (d 1)j 1 + (d 1)n = dn . N We choose N large enough so that d 1 . < d 4 With N and now xed we may choose > 0 so that for all n 1, 2 for all points a, b K such that q(a, w) < and q(b, w) < whenever w Sj (j) for j = 1, . . . , N , recalling that Sj (j) CVM = and hence we have the estimate from our choice of M . Pick u, v K such that q(u, z0 ) < /2 and q(v, z0 ) < /2 where is yet to be speci ed. Consider (5.10) dN |(T n )(a) (T n )(b)| < |(T n+N )(u) (T n+N )(v)| = d N j=1 N N (T n )(zj (u)) (T n )(zj (v)) N where zj (u) are the solutions to g(z) = u for length N words and N zj (v) is de ned similarly. We now choose small enough so that for each j = 1, . . . , N , and each point w in Sj (j) there is a solution to DYNAMICS OF RATIONAL SEMIGROUPS 55 g(z) = u and g(z) = v for some word g of length j within q-distance of w. We denote the collection of these pairs of preimages of u and v collectively by Sj (j). As the elements in Sj (j) do not arise as preimages of points in Si (i) under length (j i) words for i < j, the number may be chosen so that the same may be said of the sets Sj (j). We now divide the sum in (5.10) as follows. Recall that SN (j) consists of the preimages of Sj (j) under length (N j) words. Let SN (j) denote the preimages under all length (N j) words of the pairs of points in Sj (j) for j = 1, . . . , N and let SN (N + 1) denote the remaining preimages of u and v under length N words. The remark after the construction of the sets Sj (j) shows that the sets SN (j) are well de ned and account for all preimages of u and v under length N words. We remark that the cardinality of Sl (j) was constructed to be the same as that of Sl (j) for 1 l N and 1 j l + 1. We let SN (j) denote the sum over N (j). Note that for j = 1, . . . , N , the pairs in S SN (j) (T n )(z N (u)) (T n )(z N (v)) = d(N j) Sj (j) (T n+(N j) )(z j (u)) (T n+(N j) )(z j (v)) where z j (u) stand for solutions of g(z) = u for length j words and z j (v) is de ned similarly. Thus if u, v K, q(u, z0 ) < /2 and q(v, z0 ) < /2, then dN |(T n+N )(u) (T n+N )(v)| d N i=1 |(T n )(ziN (u)) (T n )(ziN (v))| N +1 = d N j=1 SN (j) N |(T n )(z N (u)) (T n )(z N (v))| = j=1 d j Sj (j) |(T n+(N j) )(z j (u)) (T n+(N j) )(z j (v))| |(T n )(z N (u)) (T n )(z N (v))| d 1 d j N +d N SN (N +1) N j=1 (d 1)j 1 + dj 2 2 = < 2d j=0 d 1 d + 2 56 DAVID BOYD AND RICH STANKEWITZ for all n 1. We then use the continuity of the functions T n to get the estimate |(T m )(u) (T m )(v)| < for m = 1, . . . , N , shrinking if necessary. Thus Cases 1 and 2 show that {T n } is equicontinuous at each point of K. This completes the proof of Lemma 5.2 and hence establishes the weak convergence of the measures a independently and locally n uniformly in a C \ E(G). The regularity of the limiting measure has also been established. 5.9. Proof of the inequalities (5.2). The inequalities (5.2) of Theorem 5.2 are established as follows. For any continuous real valued function on C and for a J(G), ( ) d ( ) = lim m ( ) d a ( ) = lim d m m m g(z)=a l(g)=m (z) (5.11) k k = lim d m m j=1 h(fj (z))=a l(h)=m 1 k (z) = j=1 dj lim d m+1 d m j (z) h(z)=a l(h)=m 1 = j=1 dj lim d m d k j ( ) d a ( ) = m 1 j=1 dj d j ( ) d ( ) j where j (z) = d 1 i=1 (zi,j (z)) with the zi,j (z) ranging over the j solutions of fj (w) = z. As the solutions depend continuously on z, the function j (z) is continuous for each j = 1, . . . , k. Remark 5.2. This shows that ( ) d ( ) = (T )( ) d ( ) where T is the continuous linear operator de ned on the space of continuous functions on C de ned by (5.7). Hence the measure is T invariant. For any compact set K, let the functions n be continuous and uniformly bounded, and let n decrease to k f 1 (K) as n . Thus j=1 j k (5.12) n ( ) d ( ) k j=1 1 fj (K) ( ) d ( ) = ( j=1 fj 1 (K)). DYNAMICS OF RATIONAL SEMIGROUPS 57 Recall from (5.11) that k (5.13) ( ) d ( ) = j=1 n dj d n ( ) d ( ). j We remark that for any point z C, and for any j = 1, . . . , k, dj n dj lim n (z) j = lim n d 1 j i=1 (zi,j (z)) = n d 1 j i=1 k l=1 fl 1 (K) (zi,j (z)). For any z K, and j = 1, . . . , k, as each zi,j (z) follows that n (z) decreases to K (z). Hence j (5.14) lim n ( ) d ( ) = j k l=1 fl 1 (K), it n K ( ) d ( ) = (K) K K for each j = 1, . . . , k. k 1 For z fj i=1 fi (K) \K the number counted according to multiplicity of elements in fj 1 (z) k i=1 fi 1 (K) can be any integer between k i=1 1 and dj . This integer could vary as z ranges over fj Thus for such z, 1 d j fj ( k i=1 1 fi (K))\K (z) fi 1 (K) \K. dj lim n n (z) j = d 1 j l=1 k i=1 1 fi (K) (zl,j (z)) fj ( and so 1 (5.15) fj dj k k i=1 1 fi (K))\K (z) fi 1 (K) i=1 \K k i=1 1 fi (K) lim n fj ( )\K n ( ) d ( ) j k fj fi 1 (K) i=1 \K . 58 DAVID BOYD AND RICH STANKEWITZ k i=1 k 1 1 For any z fj / i=1 fi (K) we obviously have fj (z) and so limn n (z) = 0 for j = 1, . . . , k. Hence j fi 1 (K) = (5.16) n lim C\fj ( k i=1 1 fi (K) ) n ( ) d ( ) = 0. j Together, (5.12), (5.13), (5.14), (5.15), and (5.16) yield the inequalities (5.2) for any compact subset K of C. We also have the inequalities (5.2) for any open set U . We repeat the above argument replacing K by U and replacing the functions n by continuous, uniformly bounded functions n which increase to k f 1 (U ) . Routine, but tedious, arguments, making use of the regj=1 j ularity of , may now be used to extend the inequalities (5.2) to all Borel sets. 5.10. Proof of the inequality (5.3). Now we establish the inequality (5.3) of Theorem 5.2. Let U be open and let the functions n be nonnegative, continuous on C, and increase to U . What we have shown in (5.11) is that 1 ( ) d ( ) = d n k dj n (zi,j ( )) d ( ) j=1 i=1 where the points zi,j ( ) are the solutions to fj (z) = . In particular, when fj (U ), there is at least one solution to fj (z) = in U . Hence for fj (U ), dj n lim i=1 n (zi,j ( )) fj (U ) ( ) n ( ) d ( ) k and so (U ) = 1 lim n d k dj fj (U ) i=1 U ( ) d ( ) = lim n n j=1 1 (zi,j ( )) d ( ) d 1 = d k fj (U ) ( ) d ( ) j=1 fj (U ) (fj (U )). j=1 This establishes (5.3) for open sets. Now let E be a Borel set and let the set U be open and contain E. Then (U ) 1 d k j=1 (fj (U )) 1 d k (fj (E)). j=1 DYNAMICS OF RATIONAL SEMIGROUPS 59 Taking the in mum over such sets U establishes (5.3) by the regularity of . 5.11. The Support of . We now prove the statement about the closed support of . Since the measure is independent from the initial point a, if we take a J(G) it is immediate that supp( ) J(G). Given > 0 and z J(G), let be continuous on C with 0 1 on C, 1 on (z, /2) and 0 o of (z, ). The expanding property established in Proposition 1.6 guarantees that there is an integer N such that J(G) l(g)=N g( (z, /2)). Recalling that h 1 (J(G)) J(G) for all h G, as g ranges over the words of length m+N , the equations g(w) = a for xed a J(G) have at least dm solutions in (z, /2) for any positive integer m. Thus ( (z, )) = (z, ) ( ) d ( ) lim m ( ) d ( ) = lim m a ( ) d m+N ( ) (z, 2 ) ( ) d a ( ) d N . m+N Hence supp( ) = J(G). With this, we have completed the proof of Theorem 5.2. 6. The Filled-in Julia Set for Polynomial Semigroups of Finite Type The material in this section is taken from [6]. For a polynomial f of degree at least two, the lled-in Julia set, denoted by K(f ), is de ned to be the set of points z C such that the forward orbit {f n (z)} is bounded. The complement of K(f ), the set of points which tend to under iteration of f , is called the basin of attraction of and is denoted A (f ). It is well known that K(f ) is the union of J(f ) and the bounded components of N (f ). Also, A (f ) is the component of N (f ) containing . Further, we have J(f ) = K(f ) = A (f ). (See [10], III.4.) We have the following proposition. Proposition 6.1. Let f be a polynomial of degree at least two. Then the following are equivalent: (1) A (f ) is simply connected. (2) J(f ) is connected. (3) K(f ) is connected. (4) f has no nite critical point in A (f ). For a proof see [5], Theorem 9.5.1 together with the fact that a domain is simply connected if and only if its complement is connected. 60 DAVID BOYD AND RICH STANKEWITZ Proposition 6.1 is one of the many instances where the critical points of a rational function play a strong role in its dynamics. In this chapter we point out one of the di erences in the role played by the critical points of the functions in a rational semigroup. 6.1. Polynomial Semigroups of Finite Type. Hinkkanen and Martin considered the generalization of the lled-in Julia set to more general polynomial semigroups. However, there are some questions about the proper generalization. If G is a polynomial semigroup, one need not have any point z where the set G(z) = {g(z) : g G} is bounded. See Remark 6.5 below. However, it is also possible to construct polynomial semigroups where for every point z C, the set G(z) has a nite accumulation point. To see this consider G = z 2 , z 2 /2, z 2 /3, . . . . Hinkkanen and Martin invented the concept of a polynomial semigroup of nite type as a natural compromise between the two extremes. It turns out that various one-complex-dimansional moduni spaces for discrete groups can be described as the complements of the lled-in Julia sets (de ned below) for certain polynomial semigroups of nite type. See [15] for a discussion. We summarize the de nition and main theorem on polynomial semigroups of nite type found in [15]. De nition 6.1. We say that a polynomial semigroup G is of nite type if it satis es the following conditions: (1) For any positive integer N , there are only nitely many polynomials in G whose degree is less than N . (2) There is a domain D in C, whose complement C\D is a bounded continuum, such that each g G maps D into itself, that is g(D) D. Remark 6.1. It is easy to see that every nitely generated polynomial semigroup where the generators have degree at least two is of nite type. More generally, if G is of nite type, G can only have nitely many generators of a given degree. Lastly, any degree 1 elements in G must be generated by a single elliptic M bius transformation, i.e., o must be nite order rotations around some point in C. De nition 6.2. Let G be a polynomial semigroup of nite type. The lled-in Julia set of G, denoted K(G), is the closure of the set of points z C such that G(z) = {g(z) : g G} has a nite limit point. The complement of K(G), denoted A(G) = C \ K(G), is the basin of attraction of for G. Remark 6.2. It is clear that K(g) K(G) for all g G and that g 1 (K(G)) K(G) for all g G. Either of these two statements DYNAMICS OF RATIONAL SEMIGROUPS 61 imply that J(G) K(G); the rst by Theorem 1.2 and the second by Property 1.1. The following example, provided by Hinkkanen and Martin, shows that the set A(G) need not be connected. Example 6.1. Let Dj = {z : |z aj | rj } for 1 j 3 be disks that are tangent to each other outside (with disjoint interiors). Let B be a very large disk containing all the Dj well in its interior. De ne G = g1 , g2 , g3 , where gj (z) = aj + (cj (z aj ))nj and cj > 0 is chosen so that J(gj ) = Dj while the positive integer nj 2 is so chosen 1 that for a suitable > 0, to be speci ed, we have gj (B) Bj = {z : |z aj | rj + }. Choose so small that there is still an open set W in between (in the interstice of) the disks Bj for 1 j 3. Now it is easily seen that K(G) 3 Bj . Furthermore, if z W , then any j=1 gj maps z outside B so that G(z) clusters only to in nity. So in this case the complement of K(G) has a bounded component and is not connected. Hinkkanen and Martin s main result on polynomial semigroups of nite type in [15] is the following. Proposition 6.2 ([15], Theorem 7.2). Let G be a polynomial semigroup of nite type. Then there is a domain V D, where D is as in De nition 6.1, containing a neighborhood of , such that V coincides with the unbounded component of the complement of the set g 1 g G h G K(h) , and has the following property: for any z V (and hence any z D) and for any compact subset K of C, there are only nitely many g G such that g(z) K, and, furthermore, V is the largest domain containing the point at in nity that has this property. Our rst result is that V , which may be thought of as the immediate basin of attraction for in nity, arises from simpler sets than the one in Proposition 6.2. Theorem 6.1 ([6], Theorem 5.1.7). Let G be a polynomial semigroup of nite type and let V be as in Proposition 6.2. Then V is also the unbounded component of the complements of K(G), g G K(g), and J(G). In particular, V is a component of N (G). 62 DAVID BOYD AND RICH STANKEWITZ Proof. That V is the unbounded component of A(G) = C \ K(G) was shown in the proof of Proposition 6.2. For notational simplicity let (6.1) K1 (G) = g G g 1 h G K(h) . We remark that K1 (G) is the smallest, closed set containing K(g) for all g G that is backwards invariant under each g G. Also let (6.2) We remark that (6.3) The rst inclusion follows from the fact that J(G) = g G K2 (G) = g G K(g). J(G) K2 (G) K1 (G) K(G). J(g) K(g) = K2 (G). g G See Theorem 1.2. The second inclusion of (6.3) follows from the fact that g 1 (K(g)) = K(g) for all polynomials g. The nal inclusion follows from Remark 6.2. Thus if V and V are the unbounded components of C \ K2 (G) and C \ J(G), respectively, we have the inclusions Recall that V is maximal with respect to the property that given any point z V and any compact set K C, there are only nitely many g G such that g(z) K. We wish to show that V also has this property, and hence V = V = V . The proof closely follows that of Proposition 6.2. We reproduce the relevant facts here. Recall that J(G) = g G J(g) (Theorem 1.2). For each g G of degree at least two, let Sg (z) denote the Green s function of A (g) with pole at in nity. For any z V , z lies in A (g) so Sg (z) > 0 for all g G. Further, (6.4) Sg (g(z)) = (deg g)Sg (z) for all z A (g) (see [10], p. 35.) See the same reference to establish the fact that the logarithmic capacity of J(g) satis es cap(J(g)) = 1/M 1/(n 1) where n = deg g and M is the modulus of the leading coe cient of g. Hence the logarithmic capacity of J(g) is positive. Let T (z) denote the Green s function of V with pole at in nity. As V is contained in the complement of J(g) and as both T (z) and Sg (z) V V V . DYNAMICS OF RATIONAL SEMIGROUPS 63 have logarithmic singularities at , we see that for each g G the function T (z) Sg (z) is bounded and harmonic in V , non-positive on the boundary of V and hence non-positive on V . Thus Sg (z) T (z). Hence for each z V Let N 1. By assumption, there are only nitely many g G with deg g N . Thus for z V , the numbers Sg (g(z)) = (deg g)Sg (z) (deg g)T (z) tend to as g runs over the elements of G so that the numbers deg g are non-decreasing. We wish to show that this implies that the numbers g(z) also tend to . For each g G of degree at least two, the set J(g) is compact and lies in a xed disk of radius R centered at the origin. Here R depends on V only. We know from the above that cap(J(g)) is positive, so c = log(cap(J(g))) is well de ned. There is a probability measure m on J(g) such that (6.5) Sg (z) c = J(g) inf{Sg (z) : g G} T (z) > 0. log |z w| dm(w). See [33], 1.5. Suppose that |g(z)| < r. If cap(J(g)) > L > 0, then using (6.5), we see that Sg (g(z)) = c + J(g) log |g(z) w| dm(w) < log(L) + log(R + r) since |g(z)| < r, J(g) (0, R), and m is a probability measure on J(g). If there were a positive lower bound on cap(J(g)) for g G and |g(z)| < r for in nitely many g G, we would have an upper bound on Sg (g(z)) for in nitely many g G which is a contradiction. The point z V is at a xed positive distance d from every J(g). For a given large r, consider two disjoint subsets of elements of G. First consider those g G for which cap(K(g)) > d/2 and |g(z)| < r. From the comments above, we see that there can be only nitely many such g. Next, consider those g G for which |g(z)| < r and cap(J(g)) d/2. Let g be a member of this latter set and set n = deg g and L = cap(J(g)). Then J(g) log |z w| dm(w) log d dm(w) = log d J(g) since the distance between z and J(g) is at least d, while J(g) log |g(z) w| dm(w) < log(R + r) 64 DAVID BOYD AND RICH STANKEWITZ since |g(z)| < r and J(g) (0, R). Since Sg (g(z)) = nSg (z) we see that log(R + r) log(L) > Sg (g(z)) = nSg (z) =n J(g) log |z w| dm(w) log(L) n(log(d) log(L)), hence (6.6) (6.7) (6.8) log(R + r) > n log d + (n 1)( log L) = n log 2 + log(d/2). n log d + (n 1)( log(d/2)) For a given r this last equation implies an upper bound for n = deg g, say n < n(r, d, R). By assumption, there are only nitely many g G whose degree is at most n(r, d, R), and so in this second class there are at most nitely many g G. Hence for any large r and any z V there are only nitely many g G for which |g(z)| < r. This shows that V = V and hence V = V = V . We have just shown that the sets J(G), K2 (G), K1 (G) and K(G) share the same unbounded component of their complements. We have also shown that We now show through a series of examples that the inclusions above can be strict. Example 6.2. Let G = z 2 , z 2 /a with a > 1. Then J(G) = {z : 1 |z| a} (see [15], Example 1) but K2 (G) = K(G) = {z : |z| a}. Example 6.3. Let f1 (z) = z 2 , f2 (z) = (z 10)2 +10, and let G = f1 , f2 . Note that K(f1 ) = (0, 1) and K(f2 ) = (10, 1) where (a, r) is the disk of radius r centered at a. We will show that K2 (G) is a proper subset of K1 (G). Let U = C \ (5, 9). We will show that from which it follows that g(U ) U for all g G. A straightforward calculation yields that on U , Thus by the minimum principle, f1 (U ) U . A similar calculation also yields |f2 (9ei + 5) 5| 11, hence f2 (U ) U as well. |f1 (9ei + 5) 5| 11. fj (U ) U for j = 1, 2 J(G) K2 (G) K1 (G) K(G). DYNAMICS OF RATIONAL SEMIGROUPS 65 For any g G, this shows that g(U ) U and hence U N (g) by Montel s Theorem (Proposition 1.1). In particular, U A (g) for all g G. Hence U V , where V is the domain from Proposition 6.2. Let N = 10, 1/(3 10) . We claim that N K1 (G). To see this, we can show that on N , f1 ei + 10 10 < 1 3 10 and so by the maximum principle we see that f1 (N ) K(f2 ) = (10, 1) and thus N K1 (G) by de nition. We also have that f2 (N ) U , for on N , f2 ei + 10 5 > 43. 3 10 We claim that this shows that N K(g) = for all g G. Recall that g(U ) U for all g G. Let g = gn gn 1 g1 where each gj equals f1 or f2 . If g1 = f2 , then g(N ) U since f2 (N ) U . If g1 = f1 , then g 2 (N ) U . We see this as follows. We rst show that f1 (K(f2 )) = f1 ( (10, 1)) U by calculating |f1 (ei + 10) 5| 76. Thus by the minimum principle we see that f1 (K(f2 )) U . Since k g1 = f1 , then g 2 = h f1 f2 f1 for some k 0 and some h G or h(z) = z. What we have shown above gives that k k g 2 (N ) = (h f1 f2 f1 )(N ) (h f1 f2 )(K(f2 )) = h(f1 (K(f2 ))) h(U ) U. = . In Hence N K(g) = for all g G, i.e. N g G K(g) particular, this shows that 10 K1 (G) \ K2 (G), so K2 (G) is a proper subset of K1 (G) as claimed. Example 6.4. Let G be the semigroup from Example 6.3. We construct a subsemigroup H of G such that K1 (H) is a proper subset of K(H). n Let h0 = f1 , h1 = f2 f1 , and in general let hn = f2 f1 . Let H = h0 , h1 , . . . . Note that any subsemigroup of a polynomial semigroup of nite type is itself a polynomial semigroup of nite type. Let N and U be as in the previous example. We have shown that hn (N ) (10, 1) for n = 0, 1, 2, . . . . Thus N K(H) by de nition. However we have also shown in the previous example that hn (hm (N )) U for all m, n 0 and that h(U ) U for all h H. Recall that U K(g) = for 66 DAVID BOYD AND RICH STANKEWITZ all g G. Hence if h, g H, we have g(N ) K(h) = . Thus N g 1 g H h H K(h) = and in particular, K1 (H) is a proper subset of K(H). Remark 6.3. This last example gives an in nitely generated polynomial semigroup H of nite type such that K(H) \ K1 (H) = . No such example for a nitely generated polynomial semigroup is as of yet known. 6.2. Relationship Between Critical Points and K(G). As stated in Proposition 6.1, for a polynomial f of degree at least 2, the set A (f ) is simply connected if and only if it contains no nite critical point of f . We show now through two examples that for a polynomial semigroup G of nite type, there is in general no relationship between the connectivity of the set V from Proposition 6.2 and the location of the critical points of the elements of G. Example 6.5. In this example we construct a nitely generated polynomial semigroup such that V is not simply connected, yet the nite critical points of every element g in G lie in K(G). Let G be the semigroup constructed in Example 6.3, i.e, G = f1 , f2 where f1 (z) = z 2 and f2 (z) = (z 10)2 + 10. Recall that K(f1 ) = (0, 1) and that K(f2 ) = (10, 1). By Proposition 6.2, there exists a number R > 0 so that the set G(z) = {g(z) : g G} clusters only at in nity for all z {z : |z 5| R}. Let S = {z = x + iy : |x 5| 1, |y| R + 1}. Note that S A (fj ) for j = 1, 2. Thus we may choose integers n1 , n2 1 so n that fj j (S) C \ (5, R) for j = 1, 2. n n Let G = f1 1 , f2 2 . We have shown that G (z) clusters only to in nity for z S (C \ (5, R)). Hence S (C \ (5, R)) V where V is the unbounded component of the complement of K(G ). In n particular, since K(fj j ) K(G ) for j = 1, 2, this shows that V is multiply connected. However, all of the nite critical points for elements in G lie in K(G ). The chain rule shows that any nite critical point for an element of G is a preimage of the critical points of the generators, namely 0 and 10. These two points are in K(G ) which is backwards invariant under any element of G . Hence all nite critical points of elements of G are in K(G ) as claimed. Example 6.6. For our next example, we construct a nitely generated polynomial semigroup G such that the set V from Proposition 6.2 is DYNAMICS OF RATIONAL SEMIGROUPS 67 simply connected, yet contains a nite critical point of an element of G. n Let g0 (z) = z 2 +1. As is easily seen, g0 (0) hence J(g0 ) is totally disconnected (see [10], Theorem 4.2). Further, we see that the set of purely real and purely imaginary numbers belong to N (g0 ) = A (g0 ) n as follows. For x real, |x2 + 1| = x2 + 1 > |x| and hence g0 (x) . Since N (g0 ) is completely invariant under g0 and g0 (iy) = y + 1 R for y R, we have iR N (g0 ) as well. We may nd positive numbers r1 , r2 and so that J(g0 ) is contained in the compact set C = {rei : 0 < r1 r r2 , 0 < | | } The set C may be covered by nitely many disks ( j , j ), j = 1, . . . , n, so that the union n ( j , j ) is connected and does not contain the j=1 set {z + iy : x 0, y = 0}. De ne gj (z) = (z j )2 j + j j = 0, . . . , n, choose an integer mj 1 so that gj j (D) {z : |z| > R}. m m m Let G = g0 0 , g1 1 , . . . gn n . Let V be the set from Proposition 6.2 for the semigroup G . Note that 0 D V . In particular, V contains m a nite critical point for an element from G , namely g0 0 . We now show that V , which is the unbounded component of the complement of K1 (G ) (see (6.1)), is simply connected. To do this, we need the following lemmas. Lemma 6.1 ([5], Lemma 5.7.2)). Let g be a rational function of degree d and let K be a compact connected subset of C. Then g 1 (K) has at most d components and each is mapped onto K by g. Lemma 6.2 ([6], Lemma 5.2.4). Let G = f1 , f2 , . . . , fk be a nitely generated polynomial semigroup such that the set k for j = 1, . . . , n. Note that K(gj ) = ( j , j ). Let G = g0 , g1 , . . . , gn . By Proposition 6.2 there exists a number R > 0 so that the forward orbit G(z) accumulates only at in nity for all z with |z| R. We may construct a domain D that contains 0 n such that {z : |z| R} D, yet D = . For each j=0 K(gj ) m E0 = j=1 K(fj ) 68 DAVID BOYD AND RICH STANKEWITZ is connected. Then the set K1 (G) = g G g 1 h G K(h) is also connected. Proof. Let k E1 = j=1 fj 1 (E0 ). Note that E0 E1 since K(fj ) = fj 1 (K(fj )) fj 1 (E0 ) for 1 j k. Further note that E1 is connected since each connected component of fj 1 (E0 ), of which there are only nitely many by Lemma 6.1, must meet K(fj ) and hence must meet the connected set E0 . A nite union of connected sets each meeting a given connected set such that the union contains this set must itself be connected. Hence E1 is connected. In general, de ne k Em = j=1 fj 1 (Em 1 ). As before, we can show that Em 1 Em and Em is connected for all m. Eachg set Em clearly is compact in C. Let E Em . We see that E is connected, for if there were m=0 open sets A and B such that A B = = A B and such that E A B, since each Em is connected and Em Em+1 , the set E would lie completely in A or in B. Thus E is connected. We remark that (6.9) E = g 1 (E0 ) g G since by construction, Em consists of the preimages of E0 under the length n words of G for n m. We now complete the proof that K1 (G) is connected. Let K0 (G) = g G g 1 h G K(h) , so K1 (G) = K0 (G). Note that E K0 (G). We will show that E K0 (G) is connected. From there we see that K1 (G) is connected, since K1 (G) = E K0 (G) and the closure of a connected set is connected. DYNAMICS OF RATIONAL SEMIGROUPS 69 Now for the proof that E K0 (G) is connected. First we see that J(G) E , for J(G) = g G J(g) (Theorem 1.2) and J(g) n (z) for all but at most two z C (Proposition 1.4), in particn=0 g ular for some z E0 . We then use (6.9) to conclude that J(G) E . Next we see that every component of g 1 (K(h)) meets J(G) and hence meets E for any g, h G since J(G) is backwards invariant under all elements of G. Thus the union of E and the components of g 1 (K(h)) for all g, h G, i.e., E K0 (G), is connected and hence K1 (G) is connected. Remark 6.4. Some questions remain about Lemma 6.2, namely must any of J(G), K2 (G) or K(G) be connected under the assumptions of the lemma? The construction of Example 6.6 is concluded for our semigroup m m m G = g 0 0 , g1 1 , . . . , g n n since by construction n K(gj j ) is connected and so K1 (G ) is also j=1 connected by Lemma 6.2. As V is a component of the complement of the closed, connected set K1 (G), it is simply connected. 6.3. Alternative De nitions for K(G). The following theorem, which appears in [14], provides another characterization of K(G) and relates it to the set of points whose orbit under G is bounded. Recall that G(z) = {g(z) : g G}. Proposition 6.3 (in [14]). If G is a polynomial semigroup of nite type, and if V is as in Proposition 6.2, so that V is the unbounded component of the complement of K(G), and if R > 0 is such that V contains {z : |z| > R}, then (6.10) B(G) {z : G(z) bounded} = g G m {z : |g(z)| R} is a compact set whose complement is connected. Furthermore, (6.11) B(G) K(G) = {z : |g(z)| n}. n>R N 2 g G deg g N Remark 6.5. It is often the case that B(G) = . If f (z) = z 2 , g(z) = (z 10)2 + 10 and G = f, g , then since K(f ) = (0, 1) and K(g) = (10, 1), it is easy to see that B(G) = . The following question was posed by Hinkannen and Martin. Assume there exists a number R1 > R so that if G(z) clusters to a nite 70 DAVID BOYD AND RICH STANKEWITZ point, then it clusters to a point w with |w| R1 . In this case, the characterization of K(G) in Proposition 6.3 simpli es to (6.12) K(G) = N 2 g G deg g N {z : |g(z)| R1 }. We see this as follows. Assume that z N 2 g G deg g N {z : |g(z)| R1 }. In particular, this implies that there exists a sequence of distinct elements g G such that |g(z)| R1 . Hence G(z) has a nite accumulation point, i.e., z K(G). Since K(G) is closed, we see that N 2 g G deg g N {z : |g(z)| R1 } K(G). Now assume that z is such that G(z) has a nite accumulation point. We are assuming that it must accumulate somewhere in (0, R1 ). Hence there is a sequence of elements gn G so that |g(z)| R1 . Since G is of nite type, we see that the degree of the functions gn must tend to in nity as n . Hence z N 2 g G deg g N {z : |g(z)| R1 }. Since K(G) was de ned to be the closure of such points, we see that K(G) N 2 g G deg g N {z : |g(z)| R1 } and so we have established (6.12) assuming the existence of the number R1 . Must such a number R1 always exist? When G is nitely generated, the answer is a rmative. Theorem 6.2 ([6], Theorem 5.3.3). Let G be a nitely generated polynomial semigroup where the degree of the generators is at least two. Let V be as in Proposition 6.2 and let R > 0 be such that V contains {z : |z| > R}. Then there exists a number R1 > R > 0 so that if z is any point such that G(z) has a nite cluster point, then G(z) clusters to some point w such that |w| R1 . Hence K(G) = N 2 g G deg g N {z : |g(z)| R1 }. DYNAMICS OF RATIONAL SEMIGROUPS 71 Proof. The set equality was established above, assuming the existence of the number R1 . Assume that no such number R1 exists, i.e., given any R1 > R > 0, there is a point z0 C so that G(z0 ) accumulates in C but not in the closed disk of radius R1 . Viewing the semigroup G as words in the generators {g1 , . . . , gk }, we see that there must be an integer M so that if the length of g is at least M , then |g(z0 )| > R1 > R. Let h1 , . . . , hkM be the words in G of length M in G. If G(z0 ) is to have a nite accumulation point, then G(hi (z0 )) must also have a nite accumulation point for some i, 1 i k M . To see this we simply note that the words of length at least M are given by kM i=1 G hi where G hi = {g hi : g G}, and any sequence from this collection must have an in nite subsequence from some G hi . But from our original assumption, |hi (z0 )| > R1 > R for 1 i k M , so G(hi (z0 )) accumulates only to in nity. This is a contradiction. Remark 6.6. Theorem 6.2 provides the basis for a computer algorithm for generating an approximate picture of K(G) when G is nitely generated. Namely, for a suitable number R1 and a suitable positive integer N , one colors the pixel p black if and only if for each integer 1 n N , at least one word g of length n satis es |g(p)| R1 . Remark 6.7. We have shown that if G is a nitely generated polynomial semigroup, and z is such that G(z) has a nite accumulation point, then it must have an accumulation point in a disk centered at 0 of radius R1 , where R1 is independent of z. We make the simple remark that G(z) need not have all its accumulation points in this disk. Let f (z) = z 2 and g(z) = (z 10)2 + 10. Let G = f, g . If w (10, 1), then lim g n (w) = 10. n Then for any xed k 1, n lim f k (g n (w)) = 102 , k so the accumulation points of G(w) accumulate to in nity. 7. Ahlfors Theory of Covering Surfaces Let f (z) be meromorphic on a domain . We de ne the spherical derivative by |f (z)| . f # (z) = 1 + |f (z)|2 72 DAVID BOYD AND RICH STANKEWITZ If f (z) is meromorphic in |z| r, denote 2 L(r) = |z|=r f # (z)|dz| = 0 |f (rei )|r d , 1 + |f (rei )|2 and (7.1) S(r) = 1 |z|<r 1 |f (z)|2 dx dy = (1 + |f (z)|2 )2 2 0 0 r |f ( ei )|2 d d (1 + |f ( ei )|2 )2 i.e., L(r)= the length of the image of the circle |z| = r on the Riemann sphere; S(r) = (1/ ) area of the image of the disk |z| < r on the Riemann sphere, determined with regard to multiplicity. Let us suppose f (z) is meromorphic in |z| r. Let D be a domain on C, and let I(r, D) denote the area of the image f ({|z| r}) which lies over D (with regard to multiplicity). Let I0 (D) denote the area of D. In this section all domains will be taken to be Jordan domains each of which is bounded by a sectionally analytic (s.a.) Jordan curve (see [12], p.126). Setting S(r, D) = I(r,D) (see [28], p. 29) we state I0 (D) Theorem 7.1 (First Fundamental Theorem). There is a constant h1 = h1 (D) such that |S(r) S(r, D)| h1 L(r). Furthermore, suppose is a subdomain of |z| < r, with {|z| = r} = , which is mapped by f (z) in a p-to-one fashion onto D. Then is called an island over D of multiplicity p, and in this instance, such an island contributes the quantity p to S(r, D). If p = 1, we say is a simple island. Theorem 7.2 (Second Fundamental Theorem). Let D1 , . . . , Dq , q 3, be Jordan domains on the w-sphere having disjoint closures. Then there exists a constant h2 depending only on the domains Dj such that q j=1 (S(r) n(r, Dj )) 2S(r) + h2 L(r), where n(r, D) is the total number of distinct islands over D in |z| < r without regard to multiplicity. Proof. This is almost the same statement as is Theorem 5.5 in [12]. We will translate the necessary notations. Suppose that the domain Dj is i i covered by the islands Dj for i = 1, . . . , k(j). Each island Dj is mapped by f onto Dj such that each point is covered equally often (counting i i multiplicity). Let n(Dj ) denote this multiplicity. Letting (Dj ) denote DYNAMICS OF RATIONAL SEMIGROUPS 73 i i the Euler characteristic of Dj we de ne the excess n1 (Dj ) of the island i Dj by i i i n1 (Dj ) = n(Dj ) + (Dj ). Writing n1 = (n 1)+( +1) = (n 1)+(l 1) where l(D) denotes the number of components of C \ D, we see that n1 is equal to the excess of the multiplicity of the island over 1 plus the excess of the connectivity of the island over 1. If n = 1, the map is univalent and the island is necessarily simply connected so that n1 = 0. Otherwise n1 > 0. i i Let n(Dj ) = k(j) n(Dj ) and n1 (Dj ) = k(j) n1 (Dj ). i=1 i=1 i So n1 (Dj ) n(Dj ) = k(j) (Dj ) k(j) ( 1) = k(j) = n(r, Dj ). i=1 i=1 Hence q q j=1 (S(r) n(r, Dj )) j=1 (S(r) n(Dj ) + n1 (Dj )) 2S(r) + h2 L(r) where the last inequality is the statement in Theorem 5.5 in [12]. Theorem 7.3. Let D1 , . . . , D5 be Jordan domains on the w-sphere having disjoint closures. Let f be meromorphic on the unit disc with no simple islands over any of the Dj . Then there exists an H depending only on the Dj s such that S(r) < HL(r) for all 0 r < 1. Proof. Since each island over Dj has multiplicity greater than or equal to two, S(r, Dj ) 2n(r, Dj ) for each j. By the First Fundamental Theorem (Theorem 7.1) S(r, Dj ) S(r)+ hj L(r) where hj is the constant depending only on Dj . 1 1 So n(r.Dj ) 2 S(r.Dj ) 2 (S(r) + hj L(r)) and so 5 1 n(r, Dj ) S(r) + 2 2 j=1 5 5 hj L(r). j=1 So by the Second Fundamental Theorem (Theorem 7.2), we have 1 5 5S(r) n(r, Dj )+2S(r)+hL(r) S(r)+ 2 2 j=1 and so for H = 2h + 5 j=1 5 5 hj L(r)+2S(r)+hL(r) j=1 hj we have S(r) < HL(r). 74 DAVID BOYD AND RICH STANKEWITZ Lemma 7.1. Let f be meromorphic on the unit disc such that S(r) < HL(r) for all 0 r < 1, then there exists a constant h2 depending only on H such that F # (0) < h2 . Proof. See [28], p. 84. Theorem 7.4 (Ahlfors Five Island Theorem). Let D1 , . . . , D5 be Jordan domains on the w-sphere having disjoint closures. Let f be meromorphic on the unit disc. Then there exists a constant C depending only on the domains Dj and not on f (z) such that if f # (0) > C then f (z) maps an island in the unit disc univalently onto some Dj . If |z| < R is used instead of the unit disk, then for the latter conclusion we require f # (0) > C . R Proof. The conclusion follows immediately from Theorem 7.3 and Lemma 7.1. Theorem 7.5 (Ahlfors Three Island Theorem). Let D1 , . . . , D3 be bounded Jordan domains on the w-sphere having disjoint closures. Let f be analytic on the unit disc. Then there exists a constant C depending only on the domains Dj and not on f (z) such that if f # (0) > C then f (z) maps an island in the unit disc univalently onto some Dj . If |z| < R is used instead of the unit disk, then for the latter conclusion we require f # (0) > C . R Proof. Let D4 be a Jordan domain containing that is mutually disjoint from each of D1 , D2 and D3 . Since f is analytic, there are no islands over D4 , i.e., n(r, D4 ) = 0. As in the proof of Theorem 7.3 we 1 suppose that there are no simple islands and so n(r, Dj ) 2 S(r, Dj ) 1 (S(r) + hj L(r)) for j = 1, . . . , 3. Hence by the Second Fundamental 2 Theorem (Theorem 7.2) we see that 4S(r) 3 1 n(r, Dj )+2S(r)+hL(r) S(r)+ 2 2 j=1 4 3 hj L(r)+2S(r)+hL(r) j=1 and so for H = 2h + 3 hj we have S(r) < HL(r). j=1 Lemma 7.1 can now be used to nish the proof. For similar existence of a simple island results for f (z) with regularly exhaustible Riemann surfaces see [12], p. 148 and [28], p. 30. DYNAMICS OF RATIONAL SEMIGROUPS 75 References [1] Lars V. Ahlfors. Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill, New York, 1973. [2] I. N. Baker. Repulsive xed points of entire functions. Math. Z., 104:252 256, 1968. [3] A. F. Beardon and Ch. Pommerenke. The poincare metric of plane domains. J. London Math. Soc., 18:475 483, 1978. [4] Alan F. Beardon. Symmetries of Julia sets. Bull. London Math. Soc., 22:576 582, 1990. [5] Alan F. Beardon. Iterations of Rational Functions. Springer-Verlag, New York, 1991. [6] David Boyd. Dynamics and measures for semigroups of rational functions. PhD thesis, University of Illinois, 1998. [7] David Boyd. An invariant measure for nitely generated rational semigroups. Complex Variables, To Appear. [8] David Boyd. Translation invariant Julia sets. Proc. Amer. Math. Soc., To Appear. [9] H. Brolin. Invariant sets under iteration of rational functions. Ark. Math., 6:103 144, 1965. [10] Lennart Carleson and Theodore W. Gamelin. Complex Dynamics. SpringerVerlag, New York, 1993. [11] A. Eremenko. Julia sets are uniformly perfect. Preprint, Purdue University, 1992. [12] W. K. Hayman. Meromorphic functions. Oxford University Press, 1964. [13] A. Hinkkanen. Julia sets of rational functions are uniformly perfect. Math. Proc. Camb. Phil., 113:543 559, 1993. [14] A. Hinkkanen and G.J. Martin. The dynamics of semigroups of rational functions II. Preprint. [15] A. Hinkkanen and G.J. Martin. The dynamics of semigroups of rational functions I. Proc. London Math. Soc., 3:358 384, 1996. [16] A. Hinkkanen and G.J. Martin. Julia sets of rational semigroups. Math. Z., 222(2):161 169, 1996. [17] A. Hinkkanen and G.J. Martin. Some properties of semigroups of rational functions. In XVIth Rolf Nevanlinna Colloquium (Joensuu, 1995), pages 53 58. de Gruyter, Berlin, 1996. [18] J.G Hocking and G.S. Young. Topology. Addison-Wesley, 1961. [19] A. Freire, A. Lopes and R. Ma . An invariant measure for rational maps. Bol. ne Soc. Bras. Math., 14(1):45 62, 1983. [20] M. Lyubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergod. Th. & Dynam. Sys., 3:351 385, 1983. [21] R. Ma . On the uniqueness of the maximizing measure for rational maps. ne Bol. Soc. Bras. Math., 14(1):27 43, 1983. [22] R. Ma and L. F. da Rocha. Julia sets are uniformly perfect. Proc. Amer. ne Math. Soc., 116:251 257, 1992. [23] C. McMullen. Complex Dynamics and Renormalization. Princeton Univeristy Press, 1994. [24] J. Milnor. Dynamics in One Complex Variable: Introductory Lectures. Institute for Math. Sci., SUNY Stony Brook, 1990. 76 DAVID BOYD AND RICH STANKEWITZ [25] M. H. A. Newman. Elements of the Topology of Plane sets of points. Cambridge University Press, 1939. [26] H.-O. Peitgen and P. Richter. The Beauty of Fractals. Springer-Verlag, 1986. [27] Ch. Pommerenke. Uniformly perfect sets and the poincare metric. Arch. Math., 32:192 199, 1979. [28] J. L. Schi . Normal families. Springer, New York, 1993. [29] M. Shishikura. On the quasiconformal surgery of rational functions. Ann. Sci. Ecole Norm. Sup., 20:1 29, 1987. [30] Rich Stankewitz. Completely invariant sets of normality for rational semigroups. Preprint. [31] Rich Stankewitz. Completely invariant Julia sets of rational semigroups. PhD thesis, University of Illinois, 1998. [32] Rich Stankewitz. Completely invariant Julia sets of polynomial semigroups. Proc. Amer. Math. Soc., To Appear. To appear. [33] Norbert Steinmetz. Rational Iteration: Complex Analytic Dynamical Systems. de Gruyter, Berlin, 1993. [34] Hiroki Sumi. On dynamics of hyperbolic rational semigroups. Journal of Mathematics of Kyoto University, to appear. Department of Mathematics, University of Illinois, Urbana, Illinois 61801 E-mail address: boyd@math.uiuc.edu E-mail address: stankewi@math.uiuc.edu

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