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Fractals PMA443 200809 1 Introduction to fractals
Warning: these notes are incomplete! They are intended to ensure that you have, after the lectures, an accurate written record of the details of proofs et cetera, but they contain few diagrams and no record of computer demonstrations, both of which are essential to the course. You will need to take notes which include these elements.
1.1
Administration
The course is two lectures per week, Mondays at 12.10 in Arts Tower Lecture Room 8 and Tuesdays at 3.10, in Lecture Room 9. Set work will be given approximately once a week and solutions distributed one week later; set work will be collected and marked approximately one week in three.
1.2
Books
1. K. J. Falconer, Fractal geometry: mathematical foundations and applications 2/e (Wiley, 2003), paperback: ISBN-13: 9780470848623, 33.20, ML 513.84(F), SLC 513.84(F), ASL 513(F). 2. D. Gulick,Encounters with chaos (McGraw-Hill, 1992) hardback ISBN-13: 9780070252035; paperback ISBN-13: 9780071129275. (The nearest book to the course, it is also recommended for the Chaos module PMA 324.) ML (2 OWL copies, 1 SLC) 1 HICKS SLC, all 531.3(G) (Out-of-print, but the hardback is available secondhand on Amazon from 2.95.) 3. Richard M. Crownover, Introduction to fractals and chaos (Jones and Bartlett, 1995) hardback ISBN-13: 9780867204643 41.98 (A nice introduction, but often less technical than the course.) 4. H. Lauwerier, Fractals: Images of Chaos (Penguin, 1991) ISBN-13: 9780140144116 14.00 (A popular account.) 5. B. B. Mandelbrot, The fractal geometry of nature Freeman 1983 (the original, highly eccentric work on the subject) ISBN-13: 9780716711865 45.99, ML 513.84(M). 6. M. F. Barnsley, Fractals everywhere 2/e (Academic Press,1993) 43.69 (a good systematic treatment of fractals at 3rd year undergraduate level), ISBN-13: 9780120790692, 1/e is in ML 513.84(B). 7. Y. Fisher, (ed.), Fractal image compression: theory and application, (Springer, 1995), 341pp., ISBN-13: 9783540942115, 40.99, ML B 006.42 (F) (for more technical details of the application of Iterated Function Systems to image compression). 8. H.-O. Peitgen and P. H. Richter,The Beauty of Fractals: Images of Complex Dynamical Systems (Springer, 1986) ISBN-13: 9783540158516, 31.95, (for beautiful computer graphics.) ML Q 513.84(P) 9. H.-O. Peitgen and D. Saupe The science of fractal images (Springer,1988) 23.00? (for the computer graphics enthusiast.) ML Q 513.84(S), ASL Q 513(S) 10. Manfred Schroeder Fractals, chaos and power laws: minutes from an innite paradise paperback ISBN-13: 9780716721369, 16.00 secondhand at Amazon An eccentric introduction to fractals and chaos, served with an unusually large helping of acoustic science. 11. R. L. Devaney and L. Keen Chaos and fractals: the mathematics behind the computer graphics (American Mathematical Society, Proceedings of Symposia in Applied Mathematics vol.39, 1989) ISBN-13: 9780821801376, 20.74 (A collection of articles introducing the subject; technical in places.) 1
12. K. J. Falconer, Techniques in fractal geometry (Wiley, 1997), ISBN-13: 9780471957249, 32.81, ML 513.84 (F). (A sequel to the above book by the same author; beyond the scope of this course). 13. W. A. Sutherland, Introduction to metric and topological spaces (OUP, 1975) ISBN-13: 9780198531616, 21.88 (properly includes the metric space theory referred to in the course) SLC 513.83(S)
1.3
Course Description
The concept of fractional dimension has been around for about 80 years, but the term fractal and the interest in them, both popular and scientic, date from the proliferation of microcomputers in the early 80s. The rst aim of this course is to develop an understanding of the classical theory of dimension (there are several competing denitions to consider) and its relation to the recent applications of fractals in science and technology. However fractals are not just objects with fractional dimension. Most of the well-known fractals also possess self-similarity: they are composed of several parts, each of which is a small-scale copy of the whole. Amazingly, specifying this self-similarity is enough to determine the fractal. This has important applications to the compression of image data (and was used by Microsoft in their original Encarta encyclopaedia). The proof, which is a beautiful application of the Contraction Mapping Theorem in an unusual setting, is another feature of the course. The overall structure of the course is this: we begin by studying the construction of fractals as self-similar sets; then we study dimension theory; nally, the two strands are brought together in Hutchinsons Theorem which allows one to compute the dimension from the self-similarity data for a wide class of fractals. Throughout, the chapters on fractals are interspersed with chapters on some basic analysis prerequisite to the present study and to the understanding of abstract analysis generally. Indeed, whilst the primary aim of this course is the study of dimension and self-similarity, an important secondary aim is to consolidate your knowledge of the basic ideas of abstract analysis. The sections on countability, metric spaces, compactness and Lipschitz maps should be viewed in this light. You will not be expected to use computers or to have substantial knowledge of computing.
1.4
Dimension
What exactly do we mean by the dimension of a set A R2 ? If A is a lled-in region, e.g. a disc, we want to say that A has dimension 2. If A is a straight line segment, we want A to have dimension 1. To produce a concept of dimension, we must nd a meaningful way of measuring the distinction between these two sets. There are many possibilities many notions of dimension but three stand out. We describe one of them briey here. Suppose we pick > 0 and try to cover A with -balls B(x, ). Let N () be the minimum number of such balls needed to cover A. Generally, N () as 0: but how fast? If A is a line segment, then N () 1 . If A is a disc, then N () 2 . Let us say that if N () d , then A has dimension d. More precisely, we dene KdimA = lim if the limit exists.
0
log N () , log(1/)
1.5
Fractals
Fractals occur in dynamical systems theory in many ways and they arise from geometrical constructions of self-similar sets such as the Koch curve. This curve is most easily described as the limit of a sequence of continuous curves the rst three of which are shown below. We shall see later that it has dimension log 4/ log 3. There is a growing interest in the geometry of objects such as this. The subject originated early this century, but only achieved popular recognition through the eorts of B. Mandelbrot. He coined 2
the term fractal and refused to give a formal denition for it! His book The Fractal Geometry of Nature, together with the availability of low-cost microcomputers from the late 1970s onwards, made the mathematical world, and the general public, aware of fractal geometry. His general thesis is that Euclidean geometry is based on shapes which have certain invariance properties, (a circle is invariant under rotations), but not others, (a circle is not invariant under scaling). Self-similar sets, (such as the Koch curve: one third of the Koch snowake), have invariance under scaling, but, typically, not under rotation. Thus, to a disinterested deity, self-similar sets are just as natural as circles. Indeed, these seem much more akin to many of the shapes found in the natural world. A branch of a fern, for example, has side shoots, each of which resembles the whole branch on a smaller scale. These ideas have led Barnsley to remark that a picture of a fern can be encoded into a very small amount of memory. From this, he has built a technique for encoding visual images which currently looks quite promising as a solution to the problem of compressing visual image data suciently to make video-phones viable as a replacement for the telephone. In order to discuss objects like the Koch snowake, we need to use a limiting process. With the Koch snowake, this is fairly easy, since it is a continuous curve in an obvious way. A continuous curve, in R2 , is a pair (x(t), y(t)) of continuous, real-valued functions of a real variable and limits of such functions have been well discussed in real analysis courses. However, we shall generally be looking at limits of more complicated sets. We need a theory of convergence of sets in Rn . We need a metric on a space whose points are subsets of Rn . Since we shall be using a lot of metric space theory, for which we include a brief revision chapter. Assuming this, we shall develop the theory of dimension in fairly general metric spaces rather than just in Rn . We do this, partly for notational convenience, but mainly because it shows how little of the structure of Rn we need to use, and consequently makes the arguments clearer. By way of introducing fractals in a more formal way than above, we begin by discussing the simplest fractal of allthe Cantor ternary set. This also enables us to provide a link to last semesters Chaos course, for those students attending both, by showing how it arises in connection with the dynamics of a certain unimodal function. (We shall provide sucient explanation to make the mathematics clear to those who have not attended the Chaos course, though without the motivation, they may not nd the result so exciting!)
3
2
Metric spaces (revision) and products of metric spaces (new)
distance(x, y) = |x y| (x, y R).
Basic idea: much of real analysis can be done in terms of the distance function
We abstract the properties of this function needed for analysis. Denition 2.1 A metric on a set X is a function d : X X R+ such that 1. d(x, z) d(x, y) + d(y, z) (x, y, z X),
2. d(x, y) = d(y, x) (x, y X), 3. d(x, y) = 0 i x = y (x, y X). A metric space is a pair (X, d) consisting of a set X and a metric d on X. Example 2.2 1. The real line R, with metric d(x, y) = |x y|.
2. The real plane R2 , with metric d((x1 , x2 ), (y1 , y2 )) = |x1 y1 |2 + |x2 y2 |2 .
Many of the notions associated with real analysis have generalizations to metric spaces. Denition 2.3 A sequence (xn ) in a metric space X (strictly, (X, d)) converges to a point x X i for all > 0 there exists N Z+ such that for all n N , d(xn , x) < . Denition 2.4 A subset A of a metric space X is open if, for all a A, there exists > 0 such that B(a, ) := {x X : d(x, a) < } A. Example 2.5 Every open ball B(x, ) in a metric space is an open set. Denition 2.6 A subset A of a metric space X is closed if whenever a sequence (an ) in A converges to a point x X, we have x A. Example 2.7 In R the set [0, 1] is closed, but the set (0, 1] is neither open nor closed. Typically, most subsets of a metric space are neither open nor closed. A set is closed if and only if its complement is open; consequently, the whole theory of closed sets is a reection of that of open sets, with unions becoming intersections and vice versa, subsets becoming supersets, et cetera . Denition 2.8 The closure of a set A in a metric space X is the set A := {x X : > 0 B(x, ) A = }. Equivalently, A is the smallest closed set containing A. Denition 2.9 A set D is said to be dense in a metric space X if D = X. Denition 2.10 [new] A metric space is separable if it has a countable (see next section) dense subset. Denition 2.11 If (X, d1 ), (Y, d2 ), are two metric spaces, then a function f : X Y is said to be continuous at a point x0 X i f (xn ) f (x0 ) in Y whenever xn x0 in X. If f is continuous at every point, we say that f is continuous. Denition 2.12 [new] A homeomorphism is a bijection f : X Y such that both f and f 1 are continuous. We say that X and Y are homeomorphic if there is a homeomorphism f : X Y . 4
Homeomorphisms preserve most of the important properties of metric spaces except completeness and total boundedness (see later for denitions). They preserve all properties denable in terms of convergent sequences (but not Cauchy sequences). We can prove many real analysis theorems just as easily in the context of metric spaces. For example: Proposition 2.13 In a metric space X: 1. the sets and X are open and closed; 2. the intersection of two open sets is open: the union of two closed sets is closed; 3. arbitrary unions of open sets are open: arbitrary intersections of closed sets are closed. Proposition 2.14 For a function f : X Y between metric spaces, the following are equivalent: 1. f is continuous; 2. for all x0 X and > 0 there exists > 0 such that d2 (f (x), f (x0 )) < whenever x X with d1 (x0 , x) < ; 3. f 1 (G) is open in X, for every open set G Y ; 4. f 1 (F ) is closed in X, for every closed set F Y . The proof of this depends on the Axiom of Choice, or at least its weaker version, Sequential Choice1 , to which it is equivalent. Hereafter, we shall assume this axiom without explicit mention. Curiously, many analysts are quite happy to do this but do point out places where the full Axiom of Choice is used. Denition 2.15 [new] The product of two metric spaces (X1 , d1 ) and (X2 , d2 ) is the space X1 X2 = {(x1 , x2 ) : x1 X1 , x2 X2 } with the metric d((a1 , a2 ), (b1 , b2 )) = max{d1 (a1 , b1 ), d2 (a2 , b2 )}. Exercise 2.16 Two alternative metrics on X1 X2 are the taxi-cab metric d ((a1 , a2 ), (b1 , b2 )) = d1 (a1 , b1 ) + d2 (a2 , b2 ) and d ((a1 , a2 ), (b1 , b2 )) = Show that, for a, b X1 X2 , d(a, b) d (a, b) d (a, b) 2d(a, b). d1 (a1 , b1 )2 + d2 (a2 , b2 )2 .
Denition 2.17 A sequence (xn ) in a metric space (X, d) is said to be Cauchy if, for all > 0 there exists a positive integer N such that d(xp , xq ) < for all p, q N . Informally, one may express this as d(xp , xq ) 0 as p, q . It is easy to show that every convergent sequence is Cauchy; we are particularly fond of metric spaces in which the converse holds. Denition 2.18 A metric space (X, d) is complete if every Cauchy sequence in X converges (in X). Examples 2.19 1. The set R in the usual metric is a complete metric space.
2. The set Q (the rationals) in the usual metric is not complete, because a sequence xn Q with xn 2 in R is convergent in R, so Cauchy in R, so Cauchy in Q, but is not convergent in Q.
1 Given
a sequence of non-empty sets (Xn ), there is a sequence (xn ) with xn Xn for all n.
5
3
Countability
The notions of countable and uncountable sets underlie much of the theory of dimension. For example, we shall prove the following theorem Theorem. If Ai (i = 1, 2, 3, . . .) are subsets of a metric space X, then
Hdim
i=1
Ai = sup(HdimAi ).
i
Here Hdim is a notion of dimension, to be dened later. It will be a sensible notion; for example, we shall have Hdim{point} = 0 and Hdim{line} = 1. However, this would produce a contradiction if we could write
R = {x1 , x2 , x3 , . . .} =
i=1
{xi }.
Therefore, the fact, proved below, that this cannot be done is crucial to the attempt to produce a satisfactory denition of dimension. Denition 3.1 A set C is said to be countable if either it is empty or there is a surjection : N C, (where N := {1, 2, 3, . . .}). A set is said to be uncountable if it is not countable. In other words, C is countable if it may be written C = {c1 , c2 , c3 , . . .}, possibly with repetitions. (This rewriting of the denition results from writing ci for (i).) Let us remove the repetitions, i.e. dene : N C by letting (n) = (i) with i minimal such that (i) {(1), . . . , (n 1)}. Then either is a bijection between N and C or the denition of (n) fails at some point; in which case, is a bijection between {1, 2, . . . , n 1} and C. In the former case, we say that C is countably innite; in the latter, (which includes the case C = ) that C is nite. In both cases, the inverse of gives us an injection of C into N. (In the case C = , the mapping is the empty mapping.) To summarise: Proposition 3.2 (new) For a set C, the following are equivalent: (i) C is countable; i.e. C = or there is a surjection N C; (ii) there is an injection C N; (iii) there is a bijection between C and either N or the set {1, 2, . . . , n} for some n N {0}. Warning: we say that a set C is countably innite if there is a bijection between C and N. In PMA344 the word countable was used for this. Our denition of countable is countably innite or nite. Examples 3.3 (i) The integers are countable: Z = {0, +1, 1, +2, 2, +3, . . .}. (ii) The positive rationals are countable: Q+ = 1 2 1 3 2 1 4 3 2 1 5 , , , , , , , , , , ,... , 1 1 2 1 2 3 1 2 3 4 1
with repetitions. Note that we have grouped numbers with the sum numerator + denominator equal to 2, then 3, 4, 5, et cetera. 6
(iii) The fact that the set Q of all rationals is countable is easily proved by combining the two previous ideas. One of the most useful results for proving sets countable is the following. Theorem 3.4 Every countable union of countable sets is countable. That is, if each of the set Ai is countable (i = 1, 2, 3, . . .), then A = i=1 Ai is countable. Proof. Clearly, we may assume that the Ai are all non-empty. Let A1 A2 A3 A4 = = = = ... {a11 , a12 , a13 , a14 , . . .}; {a21 , a22 , a23 , a24 , . . .}; {a31 , a32 , a33 , a34 , . . .}; {a41 , a42 , a43 , a44 , . . .};
(Note that the case of nitely many Ai is included by the simple expedient of repeating the Ai s in the enumeration.) Then A = {a11 , a21 , a12 , a13 , a22 , a31 , a41 , a32 , a23 , a14 , . . .}, (with possible repetitions). The other basic ways of inferring countability of some sets from the countability of others are contained in the following easy proposition. Proposition 3.5 (new) (i) If A is countable and f : A B is a surjection, then B is countable.
(ii) If A is countable and g : B A is an injection, then B is countable. In particular, subsets of countable sets are countable. Proof. For (i): Assume A is countable. If A = then B = . Otherwise, there is a surjection : N A so we have a surjection f : N B, showing that B is countable. For (ii): if A is countable then, by Proposition 3.2, there is an injection : A N; hence we have an injection g : B N, which proves the countability of B, by Proposition 3.2 again. The key fact that makes countability interesting is that the reals are uncountable. Theorem 3.6 Each of the sets [0, 1), R and R \ Q (the irrationals) is uncountable. Proof. We prove rst that [0, 1) is uncountable, and the rest will follow easily. Every x [0, 1) has a decimal expansion x = 0.x1 x2 x3 . . . not ending in an innite string of nines. Suppose [0, 1) is countable. Clearly [0, 1) is innite; let [0, 1) = {x(1) , x(2) , x(3) , . . .} where x(1) x(2) x(3) = = = ... We now construct a number y = 0.y1 y2 y3 . . . dierent from the above by dening yi = 0 1 if xi = 0 (i) if xi = 0.
(i)
0.x1 x2 x3 . . . , 0.x1 x2 x3 . . . , 0.x1 x2 x3 . . . ,
(3) (3) (3) (2) (2) (2)
(1) (1) (1)
Then 0.y1 y2 y3 . . . is the decimal expansion of a number y [0, 1) in a form not ending in an innite string of nines (since it has no nines at all!). Further, there is no N such that y = x(n) because 7
yn = xn , by construction. This contradicts the supposition that [0, 1) = {x(1) , x(2) , x(3) , . . .} and proves the theorem. The fact that R is uncountable follows from Proposition 3.5(ii) since if R were countable, then [0, 1), being a subset of R, would be countable too. Likewise, [0, 1] is uncountable. Finally, we know that Q is countable, so if R \ Q were countable, then R = Q (R \ Q) would be countable, by Theorem 3.4. This is not so; therefore R \ Q is uncountable. Exercise 3.7 A real number is said to be algebraic if it is a root of an equation an xn + an1 xn1 + . . . + a2 x2 + a1 x + a0 = 0, with integer coecients an , . . . , a0 , (not all zero). By considering the size of the set AN of all numbers x satisfying such an equation with |an | + . . . + |a0 | N and n N , or otherwise, show that the set of all algebraic numbers is countable. A real number is transcendental if it is not algebraic. Show that the set of all transcendental numbers is uncountable. Deduce that transcendental numbers exist! (Actually, this is the easiest way of proving the existence of transcendental numbers. It is much harder to produce a specic transcendental number and very much harder to prove that interesting numbers such as e and are transcendental.)
(n)
4
The Cantor ternary set
Denition 4.1 The Cantor Middle-Thirds Set or Cantor Ternary Set is the set C R dened as follows. Let C0 C1 C2 ... Then let C=
n=0
=
[0, 1], 1 2 = [0, ] [ , 1], 3 3 1 2 1 2 7 8 = [0, ] [ , ] [ , ] [ , 1], 9 9 3 3 9 9
Cn .
0 1/3 1/9 2/9 2/3 7/9 8/9
1
C0 = [0,1] C1 C2 C3 C4
8
Proposition 4.2 The Cantor ternary set is closed and contains no non-trivial intervals Proof. Each Cn is closed, since it is a nite union of closed intervals, so C is closed, (because every intersection of closed sets is closed.) To show that C contains no non-trivial intervals, we observe that Cn contains no intervals of length greater than 3n . If C were to contain an interval of length > 0, we should have > 3N for some N , so the interval could not be contained in CN , and so not in C. We can describe the set C in terms of the possible ternary expansions of its points. Sample ternary expansions: 25 (decimal) = 221 ternary 5/9 = 0.555 . . . (decimal) = 0.12 ternary 1/2 = 0.5 (decimal) = 0.111 . . . ternary
1 2 C1 = [0, ] [ , 1], 3 3 is the set of all numbers which have a ternary expansion starting 0.0 . . . or 0.2 . . .. Note the careful wording, due to the fact that some numbers have two possible ternary expansions: the number 1/3 is most naturally written in ternary as 0.1, but it also has the ternary expansion 0.02222 . . .; likewise 1 = 0.2222 . . .; both of these are in C1 . The set 1 2 1 2 7 8 C2 = [0, ] [ , ] [ , ] [ , 1], 9 9 3 3 9 9 is likewise the set of all numbers which have a ternary expansion of the form 0.a1 a2 . . . with a1 , a2 {0, 2}. Generally, Cn is the set of all numbers which have a ternary expansion of the form 0.a1 a2 . . . with a1 , . . . , an {0, 2}. It follows that C is the set of all numbers which have a ternary expansion of the form 0.a1 a2 . . . with an {0, 2} for all n. This representation leads on to the following proposition. Proposition 4.3 The set C is uncountable. Proof. We dene a map : C [0, 1] as follows. Let x C have a ternary expansion x = 0.x1 x2 . . . with all the xn {0, 2}, (note that such an expansion is unique), then (x) [0, 1] is dened by the binary expansion x1 x2 0. .... 2 2 Now every y [0, 1] has at least one binary expansion y = 0.y1 y2 . . . with the yi {0, 1}, so y = (x) where x = 0.a1 a2 . . ., with each ai = 2yi , is in C. Thus is surjective. Since [0,1] is uncountable, it follows that C is uncountable. (By a modication of this proof, we can nd a bijection between C and [0,1].) This result is particularly surprising if you try to get an idea of the length of C. The sets Cn , being nite unions of intervals, have clearly dened lengths. In fact the length of Cn is (2/3)n . Thus C is contained in arbitrarily short sets, and so any reasonable extension of the notion of length to encompass sets such as C (which is what is involved in the subject of Measure Theory) gives C a length of zero. Nevertheless, C is uncountable. The map constructed in this proof is quite interesting in its own right. It extends to a continuous function : R I by dening to be constant in every open interval in the complement of C. Alternatively, one may approach the denition the other way round by dening on R \ C rst: (x) (x) (x) = = = 0 1 1 2 9 (x < 0) (x > 1) (x 1 2 , ) 3 3
The set
(x) = (x) ... =
1 4 3 4
(x (x
1 2 , ) 9 9 7 8 ) , 9 9
This denes a monotonic non-decreasing function on R\C, which may then be extended to a monotonic non-decreasing function on R by dening (x) = sup{(u) : u R \ C, u < x} It is easy to see that so dened is continuous on R, since |(x) (y)| < 2n whenever |x y| < 3n (x, y R). (x C).
On [0,1], the function climbs from 0 to 1, but it is dierentiable with derivative 0 on all the intervals of I \ C. This poses severe problems for any advanced theory of integration. When you dierentiate this function, you get a function which is zero except on the set C which, having length zero, is negligible as far as integration goes. Therefore, integrating this, we get the zero function, which is not what we started with! The graph of is sometimes called a Devils Staircase. The Cantor set arises naturally in the study of the dynamical systems. We require only one denition, from the very beginning of the Chaos module. Denition 4.4 Let f : R R be any function and n any positive integer. We shall write f n (x) = f (f (. . . (f (x)) . . .));
n
that is, f 0 (x) = x and f n+1 (x) = f (f n (x)) (n = 0, 1, 2, . . .). The function f n is called the nth iterate of f . Studying the dynamical behaviour of the function f means studying the sequence of iterates (f n (x)) for various starting points x R. Consider the following function W , which I shall call the wigwam function. It is like the tent map introduced in the Chaos module, but its peak is a bit higher, (and wigwams are a bit taller than ordinary tents, arent they?) We dene W : R R by W (x) = 3x (x 3 3x (x
1 2) 1 2)
If x < 0 then W n (x) = 3n x as n . If x > 1 then W (x) < 0 and so, again, W n (x) as n . If x ( 1 , 2 ) then W (x) > 1 and so, yet again, W n (x) as n . If 3 3 1 2 7 8 x C1 \ C2 = ( , ) ( , ) 9 9 9 9 then W (x) ( 1 , 2 ) and so W n (x) as n . In fact, it is easy to see that if x Ck , then 3 3 W k+1 (x) < 0, and hence W n (x) as n . Thus the dynamical behaviour of W o C is trivial. The interesting dynamics of W are conned to C. Observe that W : Ck Ck1 (k = 0, 1, 2, . . .) so W : C C. The Cantor ternary set is one of the simplest of fractals. What we have shown here is that it arises naturally from the elementary function W as soon as we ask about dynamical behaviour. The nal remarks in this section (which are not examinable) are addressed to those who attended the Chaos module. The dynamics of W on C may be described precisely by showing that the dynamical system (W, C) is topologically conjugate to a dynamical system (, 2 ) consisting of the shift map on a space of innite sequences. (You will see references to this in past papers, but it is not now part of either module.) This topological conjugacy provides an easy proof that the dynamical system (W, C) has the following properties: 10
W
1
0
1/3
2/3
1
1. |Pern (W )| = 2n ; 2. the set Per(W ) is dense in C; 3. there is a dense orbit for W in C. The picture we have described here is very similar to that which obtains in the case of the quadratic maps F ( > 4): a set which is homeomorphic to the Cantor set away from which the dynamical behaviour is trivial and on which the dynamics are those of the shift map. A similar picture occurs for the quadratic maps Qc (c < 2), the Cantor-like set being the Julia set. Indeed, all the Qc with c outside the Mandelbrot set show the same pattern, with a Julia set homeomorphic to the Cantor set, though spread out across the plane rather than conned to a line. This is a picture which occurs frequently in dynamical systems theory.
5
Compactness
We recall the results of the nal chapter of PMA307 Metric Spaces. Denition 8.1. Let A X be a subset of a metric space. We say that A is compact if every sequence in A has a subsequence that converges to a point of A. Example 8.2. All closed intervals of the form [a, b] are compact, by the Bolzano-Weierstrass Theorem. However, the real numbers R, or any interval which is not bounded such as [a, ) or (, b], are not compact. For example, in R there is the sequence 0,1,2,3,. . . which has no convergent subsequence. Lemma 8.3. Let A X be a closed subset of a compact space X. Then A is compact. Proposition 8.4. Let A be a compact subset of a metric space X. Then A is complete and so A is closed in X . Denition 8.5. A subset A of a metric space (X, d) is bounded if there is a D > 0 such that d(a, b) D for all a, b A. Equivalently, A is bounded if A B(x, R) for some x X and R > 0. Proposition 8.6. Let A be a compact subset of a metric space (X, d). Then A is bounded. Theorem 8.7 (Heine-Borel). A subset of RN with the Euclidean metric is compact if and only if it is closed and bounded.
11
Example 5.1 Let X = R and dene a metric d on X by d(x, y) = |x y| 1 if |x y| < 1 if |x y| 1.
Then a sequence (xn ) converges to a point x in (X, d) i xn x in the usual metric on R. Likewise, (xn ) is Cauchy in X i it is Cauchy in R. Consequently, anything dened in terms of these notions is the same in (X, d) as in R. Thus (X, d) is complete, but not compact. Clearly, from its denition, (X, d) is bounded. Thus complete and bounded does not imply compact. Theorem 8.9. Let f : X Y be a continuous map between metric spaces, and let K X be compact. Then f (K) is compact. Corollary 8.10. A function f which is real-valued and continuous on a compact set K is bounded on K and attains its bounds. Denition 8.14. Let X be a metric space. A collection {Ui : i I} of subsets of X is a cover of E X, or covers E, if Ui . E
iI
If the indexing set I is a nite set then {Ui : i I} is a nite cover. If each of the Ui is an open set then the collection is an open cover. A nite collection Ui1 , . . . , Uin with i1 , . . . in I is called a nite subcover of E if E Ui1 . . . Uin . Denition 5.2 We use the term -ball to mean an open ball of radius , i.e. B(x, ) for some point x. Denition 8.15. A subset K of metric space (X, d) is totally bounded if for every > 0 there is a nite collection of -balls covering K; i.e. for every > 0, there is a nite set {x1 , x2 , . . . , xn } K such that K B(x1 , ) B(x2 , ) . . . B(xn , ). Exercise 5.3 Show that if K is totally bounded, the xi in the above denition may be taken to lie anywhere in X. (Hint: use the xi in X for /2 to get the desired xi in K for .) Proposition 8.16. Every compact metric space is totally bounded. Denition 8.17. A metric space X is said to have the HeineBorel property if every open cover of X has a nite subcover. Theorem 8.19. Let (X, d) be a metric space. The following are equivalent: (a) X is compact; (b) X is totally bounded and complete; (c) X has the HeineBorel property. A minor variation on this is the following. Theorem 5.4 A subset K of a complete metric space X is compact if and only if it is closed and totally bounded. Exercise 5.5 Show that: (a) nite unions of compact sets are compact; (b) the intersection of a closed set and a compact set is compact. Exercise 5.6 Show that every decreasing sequence K1 K2 K3 . . . of compact non-empty sets in a metric space has a non-empty intersection. Exercise 5.7 Show that if K1 X1 and K2 X2 are compact subsets of metric spaces X1 , X2 , then K1 K2 is a compact subset of the metric space X1 X2 . 12
6
Lipschitz maps and contractions
Denition 6.1 If f : X Y is a mapping between metric spaces, then we say f is Lipschitz if there is a constant such that d(f (x1 ), f (x2 )) d(x1 , x2 ) (x1 , x2 X).
The least such is called the Lipschitz constant Lip f of f . If f is not Lipschitz, we write Lip f = . If Lip f < 1, (note, strictly less than 1) we say f is a contraction. If f : X Y is a bijection such that both f and f 1 are Lipschitz, we say f is biLipschitz. If d(f (x1 ), f (x2 )) = d(x1 , x2 ) (x1 , x2 X), we say f is a similitude. Proposition 6.2 All Lipschitz maps are continuous. Proof. obvious. However, not all continuous maps are Lipschitz, and it will turn out that Lipschitz maps rather than continuous maps are the key to the theory of fractals. Example 6.3 Let f : [0, 1] [0, 1] ( with the usual metric on [0,1]) be the positive square root function. Then if d(f (0), f (x)) d(0, x) (x [0, 1]) we have so and there is no nite satisfying this. Proposition 6.4 If f : R R is dierentiable then f is Lipschitz with Lip f |f (x)| for all x R. Proof. (Exercise.) Example 6.5 The modulus function f : R R, f (x) = |x| is not dierentiable at 0, but is Lipschitz with Lipschitz constant 1. Example 6.6 Let f : Rn Rn be an ane map, i.e. f ((x1 , x2 , . . . , xn )) = (y1 , y2 , . . . , yn ) + (b1 , b2 , . . . , bn ), where yi =
j=1 n
x
x x1/2
(x [0, 1]) (x [0, 1]) if and only if
aij xj ,
for some xed matrix A = (aij ) and vector b = (b1 , b2 , . . . , bn ). In matrix notation, f (x) = Ax + b. Then d(f (x), f (y)) = =
i=1 n i=1
d(Ax, Ay)
n n
2 aij (xj yj )
j=1
n j=1
a2 ij
n j=1
(xj yj )2 (1)
=
i,j
a2 d(x, y), ij
13
where (1) uses the Cauchy-Schwarz inequality. Actually, one can do better than this: the precise Lipschitz constant of f is called the operator norm A of the matrix A, and it can be shown that it is the square root of the largest eigenvalue of AT A. Here, AT denotes the transpose of A. All the eigenvalues of AT A are real and non-negative. If A is invertible, then so is f and f 1 (x) = A1 (x b) = A1 x A1 b, so f is biLipschitz. It may be shown that, when Rn has the usual Euclidean metric, the map f is a similitude if and only if A is a scalar multiple of an orthogonal matrix. Theorem 6.7 (The Contraction Mapping Principle) Let (X, d) be a complete metric space. Let f : X X be a contraction with Lip f = [0, 1). Then f has a unique xed point x X and d(x0 , x) (1 )1 d(x0 , f (x0 )) (x0 X).
This has, essentially, been shown in the the PMA307 Metric Spaces course. What was proved was the following. Proposition 7.8. Let f : X X be a contraction of the complete metric space (X, d), so that d(f (x), f (y)) kd(x, y) for some 0 k < 1, and let x0 be any point of X. Then the sequence (xn ) dened by xn+1 = f (xn ) converges to the unique xed point x. Furthermore, for any n we have d(xn , x) kn d(x0 , f (x0 )). 1k
Thus we get a bound for the distance of xn from the limit x in terms of x0 . Writing in place of k and specializing to the case n = 0, we get Theorem 6.7. Exercise 6.8 Show that total boundedness and completeness are each preserved by biLipschitz maps. By considering the map x tan x : (/2, /2) R, show that neither is preserved by homeomorphisms. (You may assume that tan and arctan are continuous on the domains in question.)
7
The Hausdor metric
In this chapter we shall look at the set of all compact subsets of RN and dene a metric on this set, so that we shall be able to talk of a sequence (Kn ) of compact sets converging to a compact set K. Our construction would work equally well for the set of all compact subsets of a general metric space, but all our applications will be in RN , and it is more pleasant to begin our expedition into the new territories from the familiar ground of RN . Let HN denote the set of all non-empty compact subsets of RN . For x RN and K HN we dene d(x, K) = inf{d(x, y) : y K}. Note that the function y d(x, y) is a continuous function from K into R (easy exercise), so, by an exercise on compact sets, if K is compact, this function is bounded below and attains its bound; i.e. there exists y0 K with d(x, K) = d(x, y0 ) so we may write d(x, K) = min{d(x, y) : y K}. If, further, x K, then x = y0 , so d(x, K) = d(x, y0 ) > 0. Thus d(x, K) = 0 if and only if x K. Proposition 7.1 If K HN then x d(x, K) is a continuous function on RN .
14
Proof. For x, y RN and a0 K such that d(y, K) = d(y, a0 ), d(x, K) Likewise d(y, K) Combining these gives |d(x, K) d(y, K)| d(x, y). So, x d(x, K) is Lipschitz, with Lipschitz constant 1, and is therefore continuous. Again, for A, B HN we deduce sup{d(x, B) : x A} = max{d(x, B) : x A}, because the function x d(x, B) is continuous on the compact set A. We shall call this quantity (A, B). Then 1. (A, C) = max{d(a, C) : a A} = d(a0 , C), for some a0 A, d(a0 , b) + d(b, C), for all b B, by (2), d(a0 , b) + (B, C), for all b B. d(x, y) + d(x, K). d(x, a0 ) d(x, y) + d(y, a0 ) = d(x, y) + d(y, K). (2)
So (A, C) = inf{d(a0 , b) : b B} + (B, C) d(a0 , B) + (B, C) (A, B) + (B, C).
2. (A, B) = (B, A), in general. [Draw a picture of typical sets A, B RN .] 3. We have (A, A) = 0 for all A HN , since d(x, A) = 0 when x A. Conversely, if (A, B) = 0 for some A, B HN , then d(a, B) = 0 for all a A, so a B for all a A, i.e. A B. The improvement we need to is now clear. Let dH (A, B) = max{(A, B), (B, A)} (A, B HN ).
(I have adopted a dierent notation from that in Barnsleys book because I like things called d to be metrics. His d is my ; his h is my dH .) For A, B, C HN , 1. dH (A, C) = max{(A, C), (C, A)} max{(A, B) + (B, C), (C, B) + (B, A)} max{(A, B), (B, A)} + max{(B, C), (C, B)} dH (A, B) + dH (B, C).
=
2. dH (A, B) = dH (B, A), as a result of our improvement. 3. dH (A, A) = (A, A) = 0 and if dH (A, B) = 0, then (A, B) = 0 and (B, A) = 0, so A B and B A, so A = B. Thus dH is a metric on HN . 15
Denition 7.2 The metric dH is called the Hausdor metric on HN . Barnsley calls HN the space where fractals live or (less accurately) the space of fractals. Exercise 7.3 Show that it is NOT generally true that d(x, A) = dH ({x}, A) Exercise 7.4 Show that d(x, A) (x RN ; A HN ).
d(x, B) + dH (B, A) (x RN ; A, B HN ).
Exercise 7.5 (which is needed in the next chapter). Show that, for A, B, C HN , (A B, C) = max{(A, C), (B, C)} and (A, B C) Deduce that, for A, B, C, D HN , dH (A B, C D) max{dH (A, C), dH (B, D)}. min{(A, B), (A, C)}.
Theorem 7.6 The metric space HN is complete. Actually, it is true for all complete metric spaces X that the corresponding space of all non-empty compact sets with the Hausdor metric is complete, but the easy characterization of compact sets in RN greatly simplies our proof. The general proof may be found in Barnsleys book, pages 35-39, (to which one must add the fact that completeness plus total boundedness implies compactness.) Lemma 7.7 For a metric space (X, d), the following are equivalent: (i) X is complete; (ii) if (xn ) is a sequence in X such that d(xn , xn+1 ) 2(n+1) (n 1), then (xn ) is convergent. Proof of Lemma. 1. (i) (ii) If (xn ) is a sequence in X such that d(xn , xn+1 ) d(xn , xm ) 2(n+1) (n 1), then for m n,
d(xn , xn+1 ) + d(xn+1 , xn+2 ) + . . . + d(xm1 , xm ) 2(n+1) + 2(n+2) + . . . + 2m 2n ,
so (xn ) is Cauchy. The implication (i) (ii) follows. 2. (ii) (i) Suppose (ii) and let (xn ) be a Cauchy sequence in X. Then there exist n1 < n2 < . . . such that for each r, d(xp , xq ) < 2(r+1) (p, q nr ). Therefore the subsequence (xnr ) satises the hypothesis of (ii) and so converges to some r=1 x X. But a Cauchy sequence with a convergent subsequence is necessarily convergent (by PMA307 Proposition 8.4). Therefore the whole Cauchy sequence (xn ) is convergent. Proof of Theorem. Let (An ) be a sequence in HN such that dH (An , An+1 ) < 2(n+1) for all n. By the lemma, it suces to show that such sequences (An ) are convergent in HN . Let A = {x : d(x, An ) 2n for all n 16 1}.
Then A=
Bn
n=1
where Bn = {x : d(x, An )
2n }.
Now Bn is closed: if xi Bn and xi x as i , then there exist ai An with d(xi , ai ) 2n . Since An is compact, some subsequence (aij ) converges to some a An . Now xij x, so j=1 d(xij , aij ) d(x, a). Each d(xij , aij ) 2n , so d(x, a) 2n ; i.e. x Bn . Further, Bn is bounded: An is compact, so bounded, say An B(x, r); so Bn B(x, r + 2n ). Since all closed, bounded subsets of RN are compact, (this is where our great simplication occurs), we deduce that Bn is compact, for each n. Since the An are non-empty, so are the Bn . Moreover x Bn+1 d(x, An+1 ) 2(n+1) d(x, An ) d(x, An+1 ) + dH (An+1 , An ) d(x, An ) 2(n+1) + 2(n+1) = 2n x Bn by Exercise 7.5,
Thus the Bn form a decreasing sequence of compact non-empty sets, so their intersection A is compact and non-empty, by Exercise 5.6. We show that A is the limit of the sequence (An ) in HN . If a A then d(a, An ) 2n for all n n, by the denition of A, so (A, An ) 2 . Conversely, if an An , there exists an+1 An+1 with d(an , an+1 ) 2(n+1) , then an+2 An+2 with d(an+1 , an+2 ) 2(n+2) , et cetera . It follows that the sequence (an ) is Cauchy and so convergent in RN ; an a, say. Now, for all r 0, d(a, An+r ) d(a, an+r ) = lim d(am , an+r )
m m (n+r)
lim
2(n+r+1) + 2(n+r+2) + . . . + 2m .
2
Thus a Bi (i
n). Since the sequence (Bi ) is decreasing, it follows that a
i=1
Bi = A. 2n . We have shown d(An , A) 2n , for all n, so A is
Thus a A and d(an , a) 2n ; so (An , A) the limit of the sequence (An ) in HN .
8
Iterated Function Systems
Consider the following examples of self-similar fractals: the Sierpinski triangle (or Sierpinski gasket), the Koch curve and the Barnsley fern. [PICTURES: HANDOUTS] All these are examples of sets A such that
M
A=
i=1
wi (A)
for some set {wi : 1 i M } of contractions on R2 . In fact, M = 3 for the Sierpinski triangle and M = 4 for the Koch curve. The fern is trickier: here, M = 4. The map w2 takes the fern onto that part of the fern beyond the rst two branches; w3 and w4 take the fern onto the rst two branches and w1 , (in the notation of Table 3.8.3 in Barnsleys book), takes the fern, squashes it into a straight line interval and ts this in as the bottom part of the stem. Close examination reveals that the stems are composed of straight line segments, but this imperfection is easily ignored. 17
Denition 8.1 An iterated function system (IFS) on RN is a nite set W = {w1 , w2 , . . . , wM } of contractions on RN . say We that a set A is self-similar for W if
M
A=
i=1
wi (A).
(3)
We are particularly interested in non-empty compact self-similar sets: we are not interested in the fact that, in the above examples of IFSs W in R2 , the set A = R2 satises (3). Given an IFS W, we dene a map W : HN HN by
M
W (K) =
i=1
wi (K)
(K HN ).
(Since each wi is continuous, the compactness of A implies the compactness of each wi (A); hence W (A), being a nite union of compact sets, is compact.) We are looking for xed points of W . Let si = Lip wi (1 i M ) and let s = max si . Theorem 8.2 The map W : HN HN is Lipschitz with Lip W s.
Proof Consider rst the case of just one mapping w1 . If A, B HN , then (w1 (A), w1 (B)) = max{min{d(w1 (a), w1 (b)) : b B} : a A} max{min{s1 d(a, b) : b B} : a A} = s1 (A, B).
Hence dH (w1 (A), w1 (B)) s1 dH (A, B). (In fact, in this case, Lip W = s1 , as may be seen by considering the action of W on singletons.) The general case is then immediate from the following general lemma. Lemma 8.3 If i : HN HN are Lipschitz with Lip i = si (1 by
M
i
M ), then : HN HN dened
(A) =
i=1
i (A)
is Lipschitz with Lip
maxi si .
Proof. It suces to prove the M = 2 case, as the general case is then proved by an easy induction based on the M = 2 case. If A, B HN then dH ((A), (B)) = dH (1 (A) 2 (A), 1 (B) 2 (B)) max{dH (1 (A), 1 (B)), dH (2 (A), 2 (B))}, by an exercise in Ch. 7, max{s1 dH (A, B), s2 dH (A, B)} = max{s1 , s2 }dH (A, B). This completes the proof of the lemma and hence the proof of the theorem. The main result of this theorem is that W is a contraction mapping on HN . We can therefore apply the Contraction Mapping Principle to obtain a xed point for W . Theorem 8.4 Let W be an IFS on RN . Then there is a unique non-empty compact set A HN which is self-similar for W. 18
Denition 8.5 We call this set A the attractor of W. Examples 8.6 1. The Cantor ternary set is the attractor of an IFS {w0 , w1 } on R given by: w0 (x) w1 (x) So w0 (0) = w1 (0) = 0, 2/3, = x/3, = (2 + x)/3. w0 (1) = 1/3, w1 (1) = 1.
2. The Koch curve may be dened by the IFS {w1 , w2 , w3 , w4 } on R2 such that each wi is an orientation-preserving similitude with Lip wi = 1/3 and w1 (0, 0) w2 (0, 0) w3 (0, 0) w4 (0, 0) = (0, 0), = (1/3, 0), = (1/2, 1/2 3), = (2/3, 0), w1 (1, 0) w2 (1, 0) w3 (1, 0) w4 (1, 0) = (1/3, 0), = (1/2, 1/2 3), = (2/3, 0), = (1, 0).
Alternatively, the Koch curve is the attractor of an IFS {w1 , w2 } consisting of orientation-reversing similitudes with w1 (0, 0) = (0, 0), w1 (1, 0) = (1/2, 1/2 3), w2 (1, 0) = (1, 0). w2 (0, 0) = (1/2, 1/2 3), The Contraction Mapping Principle, as we stated it, yields further information as to the location of the attractor. This translates into the following result. Theorem 8.7 Barnsleys Collage Theorem. Let K HN and > 0 be given. Let W be an IFS such that
M
dH Let A be the attractor of W. Then
K,
i=1
wi (K)
< .
(4)
dH (A, K) < where, as before, s = maxi Lip wi . 19
, 1s
The proof is immediate from the Contraction Mapping Principle (Theorem 6.7), applied to the mapping W : HN HN , since (4) is the statement dH (L, W (L)) < . Exercise 8.8 (hard). Show that if an IFS W in RN has attractor A and K is a non-empty compact set such that W (K) K, then
W n (K) = A.
n=1
Let us now consider the practical business of drawing fractals. Typically, our contractions wn are ane maps and we are probably working in R2 . One algorithm for producing the attractor is to follow the proof of the Contraction Mapping Principle: start with a set A0 HN and construct the sequence W n (A0 ). Reference http://links.uwaterloo.ca/ Another approach is, in a sense, to replace the sets W n (A0 ) by probability distributions. Select a point x at random, then plot x1 = wi1 (x), x2 = wi2 (x), . . ., where i1 , i2 , . . . are selected randomly from {1, 2, . . . , M }. The simplest example of this is the construction of the Sierpinski gasket by the Chaos Game. This game is played as follows. Select a point x0 in (or near) a triangle ABC, preferably one of the vertices; select a vertex, A, B or C at random; let x1 be the mid-point between x0 and the selected vertex; continue. With a little thought, it will be seen that this corresponds to the random selection of one of three ane maps with Lipschitz constants 1/2. References Article: http://math.bu.edu/DYSYS/chaos-game/chaos-game.html Software: http://math.bu.edu/DYSYS/applets/chaos-game.html To construct (an approximation to) the attractor by this Random Iteration Algorithm, we either let the algorithm run for a while before we start plotting, or we start with a point x0 known to be in the attractor; for example a xed point of one of the wi (one of the vertices A, B, C, in the above example). So much for the reproduction of images from an IFS. How do we produce an IFS to t a given image? The Collage Theorem is the key to this. It says that if we take a set L (a leaf in some of the pictures shown), and represent it approximately as a collage of reduced copies of itself, i.e. if we write
M
L
i=1
wi (L),
(5)
for some contractions w1 , w2 , . . . , wM , then this IFS represents L fairly well. To be precise, if the error in (5) is , then the Hausdor distance between the attractor of the IFS and L will be at most (1s)1 . Note that it helps to use wi s with small Lipschitz constants. The usefulness of this is that the attractor is close to L but, rather than being a blurred version of L as a classically engineered approximation might be, it is an image with a lot of detail, hopefully having a similar texture to L. Perhaps, in this way, it is more likely to fool the brain than is a blurred image containing as much information about L? Barnsley and his company Iterated Systems Inc. have developed these ideas into a working system for turning video pictures into IFS codes and back into pictures again. This fractal compression of images was used in an early CD encyclopaedia and you might come across fractal compressed les (with le extension .FIF) elsewhere.
9
Topological dimension
The notion of topological dimension is not our main concern in this course, but a quick discussion of it (without proofs) will set the scene for the more rened notions of dimension which follow. We begin with a few remarks about the history of dimension theory. 20
Before the advent of modern set theory and topology, the word dimension was used only in a vague sense. A set or conguration was said to be n-dimensional if n was the least number of real parameters needed to describe its points. Two problems arose in the late 19th century. 1. Cantor produced a bijection between R and R2 . This bijection was highly discontinuous, so it showed that one needed to think of continuous parameterizations. 2. Peano produced a continuous surjection f : [0, 1] [0, 1] [0, 1]. Peanos example means that the continuous parameterizations will have to be homeomorphisms. But is there another weird example which will kill that idea? Can Rn and Rm be homeomorphic with m = n? This is the key question, and it is surprisingly hard. It was solved by Brouwer in 1911. The answer was no, so mathematicians could breathe again! There was hope of a sensible topological dimension theory. Brouwers proof did not produce an explicit, workable denition of dimension. The foundations of the present theory were laid by Poincar in 1912, and the formal denition is due to Brouwer in 1913. e The theory was developed by Urysohn, Menger, Hurewicz and Tumarkin in the 1920s and the denitive account (for separable metric spaces) is Hurewicz and Wallmans classic book W. Hurewicz and H. Wallman, Dimension theory, (Princeton University Press, Princeton, 1941). (Incidentally, Mengers paper containing a recursive denition of dimension in a separable metric space was submitted to Monatshefte fur Mathematik und Physik in 1922, when he was 20; according to Kasss article about Menger in Notices of the American Mathematical Society, May 1996.) More recently, research has been concentrated on extension of the theory to general topological spaces and the denition we give below is one of these more modern developments. A good modern account is Pears book. A. R. Pears, Dimension theory of general spaces, (Cambridge University Press, Cambridge, 1975) (*The concept we are about to dene is called covering dimension. There are two other competing concepts: small inductive dimension and large inductive dimension. Covering dimension and large inductive dimension are equal in all metric spaces. In separable metric spaces, all three are equal. An example of Prabir Roy (1962) shows that they do not all coincide in some non-separable metric space.*) Denition 9.1 A covering {A } of a metric space X is a family of subsets of X such that X=
A .
An open covering is a covering each of whose sets A is open. A nite covering is one with nite. A covering {B }M is said to be a renement of {A } if for each M there is some with B A . The order of a family {A } of subsets, not all empty, is the largest integer n for which there exist 1 , 2 , . . . , n+1 such that A1 . . . An+1 = . (If there is no such integer n, we say that {A } has order . A family of empty subsets has order 1.) Denition 9.2 The topological dimension topdimX of a metric space X is the least integer n such that every nite open covering of X has an open renement of order n. If there is no such n, we write topdimX = . The key idea here is that a space of dimension n should have an open covering with no more than n + 1 sets overlapping at any point. However, we might have a space which is mainly one-dimensional but with a tiny two-dimensional piece. Such a space should be reckoned as two-dimensional, but if the two-dimensional piece were covered by one set of such a covering, it would be ignored. Hence the need to insist not just on the existence of one such covering, but on the existence of such a covering rening any given covering. 21
Theorem 9.3 Topological dimension has the following properties: 1. it is integer-valued; 2. it is preserved under homeomorphisms: if f : X Y is a homeomorphism between the metric spaces X and Y , then topdimX = topdimY ; 3. topdimRn = n (n 1); topdimX; topdim(Y1 Y2 ) topdimY1 + topdimY2 + 1;
4. if Y X, then topdimY 5. if Y1 , Y2 X, then
6. if Y1 , Y2 , . . . is a sequence of closed subsets of X, then
topdim
i=1
Yi
= sup(topdimYi );
i
7. if X1 , X2 are metric spaces, then topdim(X1 X2 ) topdimX1 + topdimX2 .
The proofs are mainly non-trivial, so, as we are really interested in more rened notions of dimension, we omit them. Examples 9.4 (without proofs) 1. Finite sets have dimension zero. Note that, in each of these examples, we are looking at the set as a metric space in its own right. Thus, singletons are open sets. 2. All countable sets have dimension zero. In particular: (a) (needed later) the set X = {1/n : n = 1, 2, 3, . . .} {0} with its usual metric as a subset of R has dimension zero; (b) the set of all rationals have dimension zero. 3. The Cantor ternary set has dimension zero. 4. The set of all irrationals has dimension zero.
10
Kolmogorov dimension
Throughout this chapter, will be less than 1. This means that the quantity log(1/), which appears frequently, is positive. We now introduce the simplest concept of dimension capable of taking fractional values. The motivating idea is the following question: if S is a d-dimensional set, how much information is required to specify the position of a point in S to within ? Rather than discuss the concept of information in detail, let us rephrase the question. Imagine S covered by balls of radius so that specifying a point to within means specifying a ball to which the point belongs. What is the least number N () of balls needed to cover S? It is easy to see that if S is a d-dimensional cube in the usual sense, then N () d as 0. (This notation means that there exist numbers 0 < a b such that a N ()/d b for all suciently small .)
22
Denition 10.1 Let K be a non-empty compact subset of a metric space X. For each > 0, let N () be the minimum number of open balls of radius (-balls) centred on points of K needed to cover K. (Since K is compact, it is totally bounded by Theorem 5.4: i.e. N () is nite. The fact that K is non-empty implies that N () > 0.) Then we dene the Kolmogorov dimension of K by KdimK = lim
0
log N () , log(1/)
if this limit exists. Otherwise we say that K does not have Kolmogorov dimension. There is another way to dene Kolmogorov dimension, using limsup in place of lim; the resulting quantity is always dened, so it is not necessary to qualify all results by requiring that the sets concerned have Kolmogorov dimension. This approach was used in this course up to June 2000. It has been superseded by the conceptually simpler approach using limits. The result is that, in general, the theorems are more untidy, but there is a benet in the theorem on dimensions of products. Kolmogorov dimension is properly called capacity, but the latter term is so frequently used in a dierent way in potential theory that I prefer to avoid it. It is also known as Minkowski dimension. Example 10.2 To illustrate this denition in action, we compute the Kolmogorov dimension of the Cantor ternary set C. We recall the denition of C. Let C0 C1 C2 ... Then C=
n=0
= = =
[0, 1], 1 2 [0, ] [ , 1], 3 3 1 2 1 2 7 8 [0, ] [ , ] [ , ] [ , 1], 9 9 3 3 9 9
Cn .
Given > 0, let n be such that 3(n+1) < 2 3n , i.e. n = [log3 (1/2)], then no -ball centred on a point of C can intersect more than one interval of Cn . This is because the gaps between the intervals of Cn are all at least 3n . Now every interval of Cn contains points of C. Therefore, at least 2n such -balls are needed to cover C: i.e. N () 2n . On the other hand, we can cover Cn+1 and so C by 2n+2 -balls centred on the end points of the closed intervals of which Cn+1 is composed. Therefore N () 2n+2 . Thus n log 2 log N () (n + 2) log 2 . log 2 + (n + 1) log 3 log(1/) log 2 + n log 3 As 0, we have n , and so the outer terms tend to (log 2)/(log 3). The Sandwich Rule implies that (the limit exists and) log N () log 2 lim = , 0 log(1/) log 3 so the Cantor Ternary Set has Kolmogorov dimension, equal to (log 2)/(log 3) = 0.6309297536 . . . . Remark 10.3 There is an interesting approximation 12 log 2 = 0.6315789 . . . log 3 19 which is the basis for the equal temperament system in music. The octave, which represents a frequency ratio of 2 is divided into 12 semitones, each representing a frequency ratio of 21/12 . A pure interval of one twelfth is a frequency ratio of 3 and this is approximated by 19 semitones: 3 219/12 . 23
Taking logs:
19 log 2. 12 For more information on scales and temperament see: log 3 Manfred Schroeder, Fractals, chaos, power laws (Freeman, 1991) 99101; Alexander Wood, The physics of music (Methuen, 1962) Chapter 11. We can draw two important morals from this, both of which dierentiate Kolmogorov dimension sharply from topological dimension. Remark 10.4 The Kolmogorov dimension is not necessarily an integer. Remark 10.5 The Kolmogorov dimension is not generally invariant under homeomorphisms. This is not so immediate to prove, but it is strongly suggested by the way that the number (log 2)/(log 3) arises from the 2 and 3 involved in the geometry of C. Let us set up a slightly dierent Cantor set D by removing the middle halves of intervals: let D0 D1 D2 and D =
n=0
= = = ...
[0, 1], 1 3 [0, ] [ , 1], 4 4 1 3 1 3 13 15 [0, ] [ , ] [ , ] [ , 1], 16 16 4 4 16 16
Dn .
The calculation of dimension goes over with 3 replaced by 4 to yield KdimD = log 2 1 = . log 4 2
However, we can easily produce a homeomorphism f : D C by noting that D is the set of all points having an expansion in the quaternary scale involving only 0s and 3s, in the same way that C consists of those numbers having an expansion in the ternary scale involving only 0s and 2s. The mapping f consists of replacing 3s in quaternary expansions of points in D by 2s and calling the resulting strings ternary expansions of points in C. Alternatively, a function f : [0, 1] [0, 1] whose restriction maps D C can be dened as the limit of a sequence of monotonic increasing functions fn : [0, 1] [0, 1] which map the intervals of Dn onto the intervals of Cn and which are linear between the end-points of 1 these intervals. The sequences (fn ) and (fn ) are both uniformly convergent, so the limit f of (fn ) is a homeomorphism. Now let us consider a notion of dimension for non-empty compact sets K Rn which is equivalent to Kolmogorov dimension, but is better suited to practical estimation. Denition 10.6 We dene the grid dimension of a non-empty compact set K Rn as follows. For each > 0, we choose a grid of n orthogonal sets of parallel hyperplanes with separation . [PICTURE] We let Ng () denote the number of (closed) grid cubes containing points of K and then dene griddimK = lim if the limit exists. As it stands, this denition depends on the choice of grid for each > 0. However, the next theorem shows that this does not matter. 24
0
log Ng () , log(1/)
Theorem 10.7 If K is a non-empty compact subset of Rn then, for any choices of grids, K has grid dimension if and only if it has Kolmogorov dimension, in which case griddimK = KdimK. (Hence griddimK is independent of the choices of grids.) Proof. Let C be a closed -grid cube containing at least one point x K. Then the diameter of C, the distance from one vertex to the opposite vertex, is n , so C B(x, 2 n ) (we allow a spare factor of 2 here to allow for the case when x is a vertex, the cube being closed and the ball open the overkill is irrelevant). Thus every closed -grid cube which meets K is contained in a 2 n -ball centred on a point of K. Now K is covered by Ng () such cubes, and therefore K is covered by Ng () of these 2 n -balls centred on points of K. Therefore the least number of such balls needed to cover K is no more than Ng (). That is, N (2 n ) Ng (), for all > 0: equivalently N () Ng 1 , 2 n
for all > 0 (by replacing by /(2 n). Conversely, every open ball of radius meets no more than 3n -grid cubes, so Ng () 3n N (). We complete the proof with an argument we shall need repeatedly, and which we therefore package as a technical lemma. Lemma 10.8 (Comparison Lemma) Let A(), B() be two positive-real-valued functions on R+ and suppose that there exist positive constants 1 , 2 , 1 , 2 such that for all > 0 (a) A() (b) B() then the limit
0
1 B(1 ) and 2 A(2 ), lim log A() log(1/) log B() log(1/)
exists if and only if the limit
0
lim
exists, in which case the two limits are equal. Before proving the lemma, we observe that the lemma will complete the proof of our theorem by putting A() = N (), B() = Ng (), 1 = 1, 1 = 1/(2 n), 2 = 3n , and 2 = 1. (Remember that n is xed, so it can happily form part of the expressions for the constants 1 and 2 .) Proof of Lemma. From (b) we have, on replacing by 2 , 3 B(3 ) where 3 = 1 and 3 = 1 . 2 2 Then log 3 + log B(3 ) log(1/3 ) log(1/) log 3 log(1/) = log (3 B(3 )) log(1/) log A() log(1/) log (1 B(1 )) log(1/) log 1 + log B(1 ) log(1/1 ) A(),
= Now as 0,we have log(1/) and so
log(1/) log 1 log(1/)
.
log(1/) log i 1 log(1/) 25
(i = 1, 3),
and
log i 0 log(1/i ) LB := lim
(i = 1, 3). log B() log(1/)
Suppose
0
exists; then, as 0, we have i 0, so log B(i ) LB . log(1/i ) Hence, in the above chain of inequalities, the rst and last expressions both tend to LB . Therefore, by the Sandwich Rule, log A() LB , log(1/) as desired. This proves half of the theorem, but the other half is similar, with the rles of A and B o being reversed. To make an experimental determination of dimension, we put -grids for various over the set K and compute Ng (). We then plot log Ng () against log(1/). Typically, these points might lie approximately on a straight line; that is, there might be a relation of the form log Ng () = c + d log(1/), coming from a relation where c = log k. In this case KdimK = lim Ng () = kd log Ng () = d, log(1/) (6)
which is the slope of the graph (6). Notice that the slope of (6) gives a better approximation to the Kolmogorov dimension than taking the value of log Ng () log(1/) for the smallest value of considered, (always assuming the linear relation (6)). This is, of course, an experimental approximation to an abstract mathematical notion. We are measuring the texture of the set K only over a certain range of scales, the range of s used, whereas the mathematical idea refers to the limit as tends to zero. Nevertheless, it is a useful way of measuring texture. Examples 10.9 1. Barnsley, in his book, gives some examples of woodcuts and invites the reader to estimate their dimensions. He conjectures that individual artists produce work of characteristic dimension. 2. Over a wide range of scales, the dimension of the surface of the human lung is 2.17. (There is in article in the February 1990 issue of Scientic American on chaos and fractals in physiology. Although it doesnt go into details, it does include some pretty pictures of a latex cast of the lung and of a computer model of a similar fractal structure.) The surface of the grey matter of the brain is even more convoluted with a dimension estimated at 2.65 0.05. 3. From PHYSICS NEWS UPDATE, The American Institute of Physics Bulletin of Physics News, Number 629 March 19, 2003.
26
BLOOD VESSEL NETWORKS. A new mathematical model is leading to insights about the formation of blood vessel networks. The model, proposed by researchers from several Italian institutions (contact A. de Candia, decandia@na.infn.it, 011+39-081676805), accurately mimics vascular structures formed by cells randomly spread on a gel matrix. Chemical cues entice cells on a growing medium to migrate and aggregate into groups. Below a certain cell density, the model and related experiments show many disconnected groups are formed. Above a critical density known as the percolation limit, a spanning cluster of cells connected across large distances is formed (images at www.aip.org/mgr/png/2003/182.htm ). Exactly at the percolation threshold, such a cluster exhibits a fractal structure with a fractal dimension of about 1.9. (The fractal dimension species how much of the available space is lled. For a 2-dimensional gel plate, the surface is entirely lled at a fractal dimension of 2.) In addition, both experiment and the new model point out that the fractal dimension is dierent when the cells are observed at dierent scales. At scales of about 0.8 millimeters or less, the fractal dimension of the cell networks drops to about 1.5. The researchers speculate that the change in dimension may be indicative of the dynamics that led to the formation of the cellular networks in the rst place. The good agreement between the model and in-vitro experiments on gel growing media suggests that we may soon gain a better understanding of the formation of vascular networks in living creatures, as well as the pathological vascular formation that accompanies certain cancers and other ailments. (A. Gamba et al., Physical Review Letters, 21 March 2003) Note the somewhat inaccurate statement The fractal dimension species how much of the available space is lled. For a 2-dimensional gel plate, the surface is entirely lled at a fractal dimension of 2. However, this does illustrate the use of fractal dimension at various scales as a convenient measure of texture. Here is another way of characterizing Kolmogorov dimension, which is, in a sense, dual to the original denition; it will be very useful later. Theorem 10.10 For a non-empty compact set K in a metric space and > 0, let M () be the largest number m for which there is set {x1 , x2 , . . . , xm } K with d(xi , xj ) for all i = j. Then lim log M () log(1/)
0
exists if and only if KdimK exists, in which case they are equal. Proof. If {x1 , x2 , . . . , xm } K with d(xi , xj ) for all i = j, and m maximal, then
K B(x1 , ) B(x2 , ) . . . B(xm , ), for if x K were a point outside all the B(xi , ), then {x1 , x2 , . . . , xm , x} would be a strictly larger set with the distance between any pair of distinct elements greater than or equal to . It follows that N () M (). Conversely, if {x1 , x2 , . . . , xm } K with d(xi , xj ) for all i = j, and if K B(y1 , /2) B(y2 , /2) . . . B(yN , /2), then no two distinct xi can belong to the same B(yj , /2). Therefore, m N . Hence M () N (/2). The theorem then follows from the Comparison Lemma by putting A() = N (), B() = M (), 1 = 1, 1 = 1, 2 = 1, and 2 = 1/2. Exercise 10.11 Show that an equivalent denition of Kolmogorov dimension is obtained if N () is replaced by the number of open balls of diameter needed to cover K. We turn now to a discussion of the basic properties of Kolmogorov di...

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Metaphysics and Epistemology of Modality: Handout 923 April 2009Kripke and Bird on necessary truths1. Identities Identity-statements involving rigid designators are necessary. This follows already from the following thesis in philosophical logic

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Times and roomsBEEM109 Experimental Economics and Finance 2009Dieter Balkenborg Monday 11-13 Friday 1-2 Both in Xfi computer suite I may want to use FEELE for some Finance and Game experiments (Econport software does not run here.) FIVE lectur