Self-Similar Sets Hefei

Self-Similar Sets Hefei - Self-Similar Sets James...

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Unformatted text preview: Self-Similar Sets James Keesling Outline •  •  •  •  •  •  •  Brief History of Self-Similar Sets Iterated Function Systems Hausdorff dimension The Code Space Shift-Invariant vs Sub-Self-Similar The Boundaries of Tiles Summary Brief History •  Helge von Koch 1904 Arkiv för matematik, Astronomi, och Fysik 1 (1904), 681-702. •  Felix Hausdorff 1918 Mathematische Annalen 79 (1918), 157-179. •  Paul Lévy 1938 Journal de l École Polytechnique 1938, 227-291 Brief History •  P.A.P. Moran Additive Functions of Intervals and Hausdorff Measure, Proceedings of the Cambridge Philosophical Society 42 (1946), 15-23. •  J.E. Hutchinson Fractals and Self-Similarity, Indiana University Mathematics Journal 30 (1981), 713-747. Brief History •  Michael Barnsley Fractals Everywhere , Academic Press (1988, 2nd ed. 1993) •  Michael Barnsley and Lyman Hurd Fractal Image Compression, A.K. Peters (1993) •  Yuval Fisher (ed.) Fractal Image Compression, Springer Verlag (1995) Brief History Helge von Koch Brief History Constantin Carathéodory Brief History Felix Hausdorff Brief History Paul Lévy Brief History Michael Barnsley Brief History Robert Devaney Brief History John Hutchinson Classical Theory of Self-Similar Fractals   Iterated " " " Function System" { f1 ,…, fN } fi : R ! R n {c1 ,…, cN } n fi 0 < ci < 1 d ( fi ( x ), fi ( y)) = ci d ( x, y) Classical Theory of Self-Similar Fractals •  Set Mapping N F ( A ) = ! fi ( A ) i =1 F : ￿(R ) ! ￿(R ) n n Classical Theory of Self-Similar Fractals n •  Contraction Mapping on ￿ (￿ ) N •  Invariant Set K = F (K ) = fi ( K ) ! i =1 The Invariant Set K 3 F (K) F( K ) 4 2 F (K) F (K ) Calculating the Hausdorff Dimension Three principles of measure: 1.  If n > m, then 2.  If m > n, then mn ( Am ) = 0 mn ( Am ) = ! 3.  If A is rescaled by a factor of t, then mn (t ! A) = t mn ( A) n Calculating the Hausdorff Dimension a!0 ! a ! >0 A! X ￿ ( A ) = inf mesh￿<! # ( diamU ) Ui "￿ ￿ a ( A ) = lim ￿ !a ( A ) ! !0 i a Calculating the Hausdorff Dimension ￿ a ( A) = ! ￿ a ( A) ￿ a* ( A ) ￿ a ( A) = 0 a* = dim ￿ A a Calculating the Hausdorff Dimension 4 "4 % m! ( K ) = m! $ ! fi ( K )' = ( m! ( fi ( K )) # i =1 & i =1 4 = " ( ) m! ( K ) i =1 1! 3 4 !( ) = 1 log 4 "= log 3 1" 3 Classical Theory of Self-Similar Fractals •  Open Set Condition There is a nonempty bounded open set O such that (1) fi (O) ! O (2) fi (O) ! f j ( O) = " Computing the Hausdorff dimension N "c i =1 ! i =1 dim H K = log 4 log 3 dim H K = log 8 log 3 Classical Theory of Self-Similar Fractals " " dim H K = log 5 log 3 Classical Theory of Self-Similar Fractals " " IFSLab Fractal Program IFSLab Fractal Program Barnsley Fern http://www.pha.jhu.edu/~ldb/seminar/images/fern.gif More Self-Similar Fractals http://www.math.sunysb.edu/~jack/CMPLX/index.html Higher Dimensional Fractals •  Menger Curve log 20 dim M = log 3 Higher Dimensional Fractals •  Sierpinski Pyramid log 5 dim P = log 2 Higher Dimensional Fractals •  Sierpinski Pyramid log 4 dim P = log 2 http://www.cs.sunysb.edu/~george/CSE391projects.htm Code Space •  Code space " ! = # {1,…, N} i =1 ! ((ij ), (k j )) = ci1 ci2 !cim g:!" K g((i j )) = lim fi1 ! " ! fim ( K ) m! " The Code Space g(1, 4, 4, 4,!) g(2,1,1,1,!) The Code Space g(5,1,1,1,!) g( 4, 7, 7, 7,!) g(6, 3, 3, 3,!) Code Space •  Dimension of the code space N "c i =1 ! i =1 dim H ! = " ! ￿ ( ") = 1 Code Space •  Similitudes on the code space fi (i1 , i2 ,…) = (i, i1 , i2 ,…) g !"K fi fi # # ! g " K Code Space •  Shift map ! :"# " ! (i1 , i2 ,…) = (i2 , i3 ,…) g N ! ( A) # ! fi ( A) "1 i =1 !"K $ %1 F # # ! g " K Code Space •  Ergodic theorem ! # A # g " g " K # E # N $ ( A) " F( E ) = ! fi ( E) %1 g i =1 Sub-Self-Similar Sets •  Sub-self-similar sets are shift invariant k E ! ! fi ( E ) i =1 ! # g "K # g $ ( A) % A " E Sub-Self-Similar Sets •  Dimension of sub-self-similar sets [Kenneth Falconer Trans. Amer. Math.Soc. 347 (1995), 3121-3129] g ! ( A) " A # E 1 & )k s ! (s) = lim( % cI + k "# ' I$Ak * dim H E = s where ! (s) = 1 Self-Similar Tilings •  Iterated Function System in ￿ { f1 ,…, fN } •  Open Sets Condition N n dim H K = n !c i =1 i =1 n Dimension of the Boundary •  A simple example Dimension of the Boundary •  Dimension O L R B dim H !K = s where " ( s) = 1 OLRB O! 3 #1 L # R# 1 # B" 0 2 2 2$ & 305 & 0 3 5& & 0 0 9% Dimension of Example •  Eigenvalues ! 3 2 2$ # 1 3 0& # & # " 1 0 3% {!1 , !2 , !3 } = {5, 3,1} ln 5 s = dim H !K = ln 3 Block Tiles ( j1 , k1 ) ( j2 , k2 ) Lévy Dragon P. Duval and J.K., Geometry and Topology in Dynamics, AMS Contemporary Mathematics Series, M. Barge and K. Kuperberg, editors, Vol 246 (1999), pp. 87-98 Lévy Dragon K0 3 F ( K0 ) F( K0 ) = f1 ( K0 ) ! f2 ( K0 ) Lévy Dragon Larry Riddle http://ecademy.agnesscott.edu/~lriddle/ifs/levy/tiling.htm Boundary of the Lévy Dragon •  Matrix M215 ! 215 M752 ! 752 M734 ! 734 ! = 1.954776399 dim H ! K = 1.934007183 Self-Similar Digit Tile •  Method for computing the dimension of the boundary of a self-similar digit tile Mathematica Program P. Duval, J. K., and A. Vince, J. London Math. Soc. (2) 61 (2000), 749-760. Self-Similar Digit Tile •  Similarity expansion matrix with integer entries " 3 !4 0% $ 4 3 0' A=$ ' $ # 0 0 5& d ￿ d •  Set of coset representatives for A￿ digit set D = {d1 , d2 , …, ddet A } The Conditions (1) lim #Tn = #T n! " (2) lim #Tn is not space filling n! " (3) {T + x x ! ￿ (4) !d T = 1 d } is a tiling of ￿ d Iterated Function System •  The IFS for a self-similar digit tile det A !1 { fi ( x ) = A ( x + di )}i =1 •  Examples "1 !1% A=$ ' #1 1 & '! 0$ ! 1$ * D = (# & , # & + )" 0% " 0% , Twin Dragon Mark McClure" The Invariant Neighborhood N0 = {0, ± e1 ,…, ± ed } N + D ! AN + D (! 0$ ! '1$ ! '1$ ! 0 $ ! 0$ ! 1 $ ! 1$ + N = )# & , # & , # & , # & , # & , # & , # & , *" 0% " 0 % " 1 % " '1% " 1% " '1% " 0% - Contact Matrix !2 #1 # #0 # C = #0 #0 # #0 # "1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 2 0 0 0 2 0 1 0 0 0 0 0 0 0 0 1 0$ & 0 & 0& & 0 & 1& & 0 & 0% Contact Matrix x !N d !D d + x ! Axd + D C = (cxy ) x ,y!N cxy = card({d xd = y}) Dimension of the Boundary # 3 3 87 + 28 1 1& log% +3 +( $ 3 3 " 3 87 + 28 3' dim H !K = log 2 = 1.523627 The Duvall Rocket ! 3 0$ A=# & " 0 3% (! 0$ !1$ ! 2$ ! '1$ ! '2$ D = )# & , # & , # & , # & , # & , *" 0% "1% " 2% " 0 % " 0 % ! '1$ ! 0 $ ! 0 $ ! 1 $ + # & , # &, # & , # & , " 1 % " '1% " '2% " '1% - The Duvall Rocket Dimension of the Boundary log(3 + 2 2 ) dim H !K = = 1.604522 log 3 Vince Gasket •  Integer similarity expansion matrix ! 2 0$ A=# & " 0 2% •  Digit set (! 0$ ! 1$ ! 0$ ! '1$ + D = )# & , # & , # & , # & , *" 0% " 0% " 1% " '1% - Vince Gasket Mark McClure" Dimension of Boundary log 3 dim H !K = = 1.5849625 log 2 Keesling Moon Lander ! 3 0$ A=# & " 0 3% '! 0$ !1$ ! 1 $ ! 2$ ! *1$ D = (# & ,# & , # & , # & , # & , )" 0% "1% " *1% " 0% " *2% ! 3 $ ! *1$ ! 3$ ! 1$ + # & , # &, # & ,# & , " *2% " 2 % " 2% " 3% - Keesling Moon Lander Dimension of Boundary 7 3 + 88 ! 3 2 3 ! 776 + 24 ! 3687i 3 + 3 776 + 24 ! 3687i ! 3 4 6 log(8.185229) dim H !K = = 1.913624 log(3) Isosceles Right Triangle Tiles Lévy Dragon Lévy Dragon Isosceles Right Triangle Tiles Isosceles Right Triangle Tiles Twindragon" Heighway Dragon" Isosceles Right Triangle Tiles Twindragon" Heighway Dragon" Tame Twindragon Two Similitudes •  Ngai, Sze-Man; Sirvent, V.F.; Veerman, J.J.P.; Wang, Yang, On 2-reptiles in the plane, Geometriae Dedicata 82 (2000), 325-244. •  Assuming rational rotations •  (! ," ) 2-reptile ( ! , 0) 4 twin dragon (! , ! ) 42 Lévy dragon ( ! , 32! ) Heighway dragon 4 ( 34! , 32! ) triangle ! ( 2 , 0 ) rectangle (tan !1 ( 7 , 0 ) tame twin dragon Triangle Tiles Triangle Tiles Triangle Tiles Triangle Tiles Triangle Tiles Triangle Tiles Triangle Tiles Subshifts of Finite Type •  The code space revisited # ! = " = $ {1,…, N} i =1 {c1 ,…, cN } 0 < ci < 1 ! ((ij ), (k j )) = ci1 ci2 !cim ! :"# " ! (i1 , i2 ,…) = (i2 , i3 ,…) Subshifts of Finite Type •  Dimension of the code space N "c i =1 ! i =1 dim H ! = " ! ￿ ( ") = 1 Subshifts of Finite Type •  Embedding the Code Space in ! i c <1 2 ￿ N e2 fi : ￿ ! ￿ N N ! i fi ( x ) = c " ( x # ei ) + ei g:!" K dim H ! = " dim H K e1 Subshifts of Finite Type •  Definition of Subshifts of Finite Type ! a11 ! a1 N $ # & A=# " # " & & # " aN1 ! aNN % { aij '{0,1} } ! A = (ij ) " # ai ji j +1 = 1 for all j ! (" A ) = " A Subshifts of Finite Type •  Dimension of Subshift of Finite Type !a c # s s s #a c (c1 , c2 ,…, cN )# " # s " aN1c1 s 11 1 s 21 1 s 12 2 s 22 2 ac ac " s aN 2 c2 ! ac$ & ! a c& # "& s& ! aNN cN % s 1N N s 2N N k '1 ! 1$ # 1& #& # 1& #& " 1% Subshifts of Finite Type •  Eigenvalues ! a11c1s # s # a21c1 M (s) = #" # s " aN1c1 s 12 2 s 22 2 ac ac " s aN 2 c2 ! ac$ & ! a c& # "& s& ! aNN cN % s 1N N s 2N N ! (s) = 1 if and only if " max (s) = 1 Example •  Well-known subshift N =2 {c1 , c2 } = {3 , 4} 55 ' ' ! 3$ ! 4$ # & +# & =1 " 5% " 5% dim H ! = 2 !1 1$ A=# & "1 0% Example •  Eigenvalues associated with subshift !( ) M (s) = # "( ) 3s 5 3s 5 ()$ 4s 5 & 0% 1 " 3% 1 " 3% " 3% " 4% $ ' + 4$ ' $ ' ! (s) = $ ' ± # 5& # 5& 2 # 5& 2 # 5& s 2s s s Example •  Dimension of subshift space ! (s) = 1 if and only if " max (s) = 1 1 " 3% 1 " 3% !max (s) = $ ' + $ ' 2 # 5& 2 # 5& s 2s " 3% " 4% + 4$ ' $ ' = 1 # 5& # 5& dim H ! A = 1.126326177 s s Summary •  Self-Similar Sets and the Code Space Probability theory Ergodic theory Information theory Dynamical systems theory •  Self-similar Tiles Generating such sets Dimension of the boundary Applications to subshifts of finite type ...
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