Modified singular value functions and self-affine carpets Jonathan - PowerPoint PPT Presentation

Modified singular value functions and self-affine carpets Jonathan M. Fraser The University of St Andrews, Scotland Jonathan M. Fraser Self-affine carpets

Self-affine sets Given an iterated function system (IFS) consisting of contracting affine maps, { A i + t i } m i =1 , where the A i are linear contractions and the t i are translation vectors, it is well-known that there exists a unique non-empty compact set F satisfying m � F = S i ( F ) i =1 which is termed the self-affine attractor of the IFS. Jonathan M. Fraser Self-affine carpets

Self-affine sets Jonathan M. Fraser Self-affine carpets

The singular value function The singular values of a linear map, A : R n → R n , are the positive square roots of the eigenvalues of A T A . For s ∈ [0 , n ] define the singular value function φ s ( A ) by φ s ( A ) = α 1 α 2 . . . α ⌈ s ⌉− 1 α s −⌈ s ⌉ +1 ⌈ s ⌉ where α 1 � . . . � α n are the singular values of A . Returning to our IFS, let I k denote the set of all sequences ( i 1 , . . . , i k ), where each i j ∈ { 1 , . . . , m } , and let d ( A 1 , . . . , A m ) = s be the solution of � 1 / k � � φ s ( A i 1 ◦ · · · ◦ A i k ) lim = 1 . k →∞ I k This number is called the affinity dimension of the attractor, F . Jonathan M. Fraser Self-affine carpets

Falconer’s theorem Theorem Let A 1 , . . . , A m be contracting linear self-maps on R n with Lipschitz � � m i =1 L n � constants strictly less than 1 / 2 . Then, for -almost all ( t 1 , . . . , t m ) ∈ × m i =1 R n , the unique non-empty compact set F satisfying m � F = ( A i + t i )( F ) i =1 has � � �� dim B F = dim P F = dim H F = min n , d A 1 , . . . , A m . In fact, the initial proof required that the Lipschitz constants be strictly less than 1 / 3 but this was relaxed to 1 / 2 by Solomyak who also observed that 1 / 2 is the optimal constant. Jonathan M. Fraser Self-affine carpets

Exceptional constructions Jonathan M. Fraser Self-affine carpets

Exceptional constructions Jonathan M. Fraser Self-affine carpets

Exceptional constructions Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets We call a self-affine set box-like if it is the attractor of an IFS consisting of contracting affine self-maps on [0 , 1] 2 , each of which maps [0 , 1] 2 to a rectangle with sides parallel to the axes. The affine maps which make up such an IFS are necessarily of the form S = T ◦ L + t , where T is a contracting linear map of the form � a � 0 T = 0 b for some a , b ∈ (0 , 1); L is an isometry of [0 , 1] 2 (i.e., a member of D 4 ); and t ∈ R 2 is a translation vector. Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Let π 1 , π 2 : R 2 → R be defined by π 1 ( x , y ) = x and π 2 ( x , y ) = y . (1) π 1 ( F ) and π 2 ( F ) are either self-similar sets or they are a pair of graph-directed self-similar sets. This shows that the box dimensions of π 1 ( F ) and π 2 ( F ) always exist and are equal in the graph-directed case. Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Let π 1 , π 2 : R 2 → R be defined by π 1 ( x , y ) = x and π 2 ( x , y ) = y . (1) π 1 ( F ) and π 2 ( F ) are either self-similar sets or they are a pair of graph-directed self-similar sets. This shows that the box dimensions of π 1 ( F ) and π 2 ( F ) always exist and are equal in the graph-directed case. Let s 1 = dim B π 1 ( F ) and s 2 = dim B π 2 ( F ) . Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Let π 1 , π 2 : R 2 → R be defined by π 1 ( x , y ) = x and π 2 ( x , y ) = y . (1) π 1 ( F ) and π 2 ( F ) are either self-similar sets or they are a pair of graph-directed self-similar sets. This shows that the box dimensions of π 1 ( F ) and π 2 ( F ) always exist and are equal in the graph-directed case. Let s 1 = dim B π 1 ( F ) and s 2 = dim B π 2 ( F ) . (2) We can compute the exact value of s 1 and s 2 in many cases. Jonathan M. Fraser Self-affine carpets

Box-like self-affine sets Let π 1 , π 2 : R 2 → R be defined by π 1 ( x , y ) = x and π 2 ( x , y ) = y . (1) π 1 ( F ) and π 2 ( F ) are either self-similar sets or they are a pair of graph-directed self-similar sets. This shows that the box dimensions of π 1 ( F ) and π 2 ( F ) always exist and are equal in the graph-directed case. Let s 1 = dim B π 1 ( F ) and s 2 = dim B π 2 ( F ) . (2) We can compute the exact value of s 1 and s 2 in many cases. (3) Compositions of maps in our IFS also map [0 , 1] 2 to a rectangle, and the singular values are just the lengths of the sides of the rectangle. Jonathan M. Fraser Self-affine carpets

Modified singular value functions For i ∈ I k let s ( i ) be the box dimension of the projection of S i ( F ) onto � [0 , 1] 2 � the longest side of the rectangle S i and note that this is always equal to either s 1 or s 2 . Jonathan M. Fraser Self-affine carpets

Modified singular value functions For i ∈ I k let s ( i ) be the box dimension of the projection of S i ( F ) onto � [0 , 1] 2 � the longest side of the rectangle S i and note that this is always equal to either s 1 or s 2 . For s � 0 and i ∈ I ∗ , we define the modified singular value function , ψ s , of S i by = α 1 ( i ) s ( i ) α 2 ( i ) s − s ( i ) , ψ s � � S i and for s � 0 and k ∈ N , we define a number Ψ s k by Ψ s � ψ s ( S i ) k = i ∈I k Jonathan M. Fraser Self-affine carpets

Properites of ψ s and Ψ s k For s � 0 and i , j ∈ I ∗ we have (1) If s < s 1 + s 2 , then ψ s ( S i ◦ S j ) � ψ s ( S i ) ψ s ( S j ) (2) If s = s 1 + s 2 , then ψ s ( S i ◦ S j ) = ψ s ( S i ) ψ s ( S j ) (3) If s > s 1 + s 2 , then ψ s ( S i ◦ S j ) � ψ s ( S i ) ψ s ( S j ) Jonathan M. Fraser Self-affine carpets

Properites of ψ s and Ψ s k For s � 0 and i , j ∈ I ∗ we have (1) If s < s 1 + s 2 , then ψ s ( S i ◦ S j ) � ψ s ( S i ) ψ s ( S j ) (2) If s = s 1 + s 2 , then ψ s ( S i ◦ S j ) = ψ s ( S i ) ψ s ( S j ) (3) If s > s 1 + s 2 , then ψ s ( S i ◦ S j ) � ψ s ( S i ) ψ s ( S j ) For s � 0 and k , l ∈ N we have (4) If s < s 1 + s 2 , then Ψ s k + l � Ψ s k Ψ s l (5) If s = s 1 + s 2 , then Ψ s k + l = Ψ s k Ψ s l (6) If s > s 1 + s 2 , then Ψ s k + l � Ψ s k Ψ s l Jonathan M. Fraser Self-affine carpets

Properites of ψ s and Ψ s k For s � 0 and i , j ∈ I ∗ we have (1) If s < s 1 + s 2 , then ψ s ( S i ◦ S j ) � ψ s ( S i ) ψ s ( S j ) (2) If s = s 1 + s 2 , then ψ s ( S i ◦ S j ) = ψ s ( S i ) ψ s ( S j ) (3) If s > s 1 + s 2 , then ψ s ( S i ◦ S j ) � ψ s ( S i ) ψ s ( S j ) For s � 0 and k , l ∈ N we have (4) If s < s 1 + s 2 , then Ψ s k + l � Ψ s k Ψ s l (5) If s = s 1 + s 2 , then Ψ s k + l = Ψ s k Ψ s l (6) If s > s 1 + s 2 , then Ψ s k + l � Ψ s k Ψ s l It follows by standard properties of sub- and super-multiplicative sequences that we may define a function P : [0 , ∞ ) → [0 , ∞ ) by: k ) 1 / k k →∞ (Ψ s P ( s ) = lim Jonathan M. Fraser Self-affine carpets

Properties of our ‘pressure’ function P P is the exponential of the function P ∗ ( s ) = lim k log Ψ s 1 k k →∞ which one might call the topological pressure of the system. (1) For all s , t � 0 we have α s min P ( t ) � P ( s + t ) � α s max P ( t ) (2) P is continuous and strictly decreasing on [0 , ∞ ) (3) There is a unique value s � 0 for which P ( s ) = 1 Jonathan M. Fraser Self-affine carpets

Dimension result Definition An IFS { S i } m i =1 satisfies the rectangular open set condition (ROSC) if there exists a non-empty open rectangle, R = ( a , b ) × ( c , d ) ⊂ R 2 , such that { S i ( R ) } m i =1 are pairwise disjoint subsets of R. Jonathan M. Fraser Self-affine carpets

Dimension result Definition An IFS { S i } m i =1 satisfies the rectangular open set condition (ROSC) if there exists a non-empty open rectangle, R = ( a , b ) × ( c , d ) ⊂ R 2 , such that { S i ( R ) } m i =1 are pairwise disjoint subsets of R. Theorem Let F be a box-like self-affine set. Then dim P F = dim B F � s where s � 0 is the unique solution of P ( s ) = 1 . Furthermore, if the ROSC is satisfied, then dim P F = dim B F = s. Jonathan M. Fraser Self-affine carpets

Some discussion (1) If s 1 = s 2 = 1, then the singular value function and our modified singular value function coincide and therefore and the solution of P ( s ) = 1 is the affinity dimension. Jonathan M. Fraser Self-affine carpets

Some discussion (1) If s 1 = s 2 = 1, then the singular value function and our modified singular value function coincide and therefore and the solution of P ( s ) = 1 is the affinity dimension. (2) The converse of (1) is clearly not true. Jonathan M. Fraser Self-affine carpets

Some discussion (1) If s 1 = s 2 = 1, then the singular value function and our modified singular value function coincide and therefore and the solution of P ( s ) = 1 is the affinity dimension. (2) The converse of (1) is clearly not true. (3) We have extended the class of self-affine sets for which it is known that the box dimension exists. Jonathan M. Fraser Self-affine carpets