Infinite iterated function systems with overlaps Sze-Man Ngai - PowerPoint PPT Presentation

Introduction WSC and topological pressures Main results Problems Infinite iterated function systems with overlaps Sze-Man Ngai Georgia Southern University and Hunan Normal University International Conference on Advances in Fractals and Related Topics The Chinese University of Hong Kong December 14, 2012 Joint with Jixi Tong, Hunan Normal University

Introduction WSC and topological pressures Main results Problems IFSs and limit set: • ∅ � = X ⊂ R d compact • I finite or countably infinite index set • { S i } i ∈ I an iterated function system (IFS) if S i : X → X are injective contractions that satisfy the uniform contractivity condition : ∃ 0 < ρ < 1 such that | S i ( x ) − S i ( y ) | ≤ ρ | x − y | ∀ i ∈ I and x , y ∈ X . • Limit set: ∞ ∞ � � � � S i | n ( X ) ⊆ K := S i ( X ) . ( K is Souslin) i ∈ I ∞ n =1 n =1 i ∈ I n

Introduction WSC and topological pressures Main results Problems • c.f. attractor or fixed point: F = � i ∈ I S i ( F ). • K satisfies � K = S i ( K ) , i ∈ I but K is not the unique set satisfying this equality, unless K is compact. Problem: Compute dim H ( K ).

Introduction WSC and topological pressures Main results Problems Motivations for studying IIFSs Fernau (1994): IIFSs have strictly more powerful descriptive power than FIFSs: • In a separable metric space, every closed set is a fixed point of an IIFS and, • there is a closed and bounded subset of a complete metric space that is a fixed point of an IIFS but not of any FIFS.

Introduction WSC and topological pressures Main results Problems Conformal IIFS Definition IFS of injective C 1 conformal contractions: if each S i can be extended to a C 1 injective conformal contraction on some bounded open connected neighborhood V of X and x ∈ V � S ′ � S ′ 0 < inf i ( x ) � ≤ sup i ( x ) � < 1 for all i ∈ I . x ∈ V Define ∞ � ∀ i ∈ I ∗ := x ∈ V � S ′ � S ′ I n . i ( x ) � , i ( x ) � , r i := inf R i := sup x ∈ V n =0

Introduction WSC and topological pressures Main results Problems Bounded distortion property Definition Bounded distortion property (BDP): ∃ c 1 > 0 such � S ′ i ( x ) � ∀ i ∈ I ∗ and x , y ∈ V . i ( y ) � ≤ c 1 � S ′ In particular, ∀ i ∈ I ∗ . r i ≤ R i ≤ c 1 r i A sufficient condition for BDP: ∃ constants C ≥ 1 and α > 0 s.t. � � � � � � S ′ i ( y ) � − � S ′ � ≤ C � ( S ′ i ) − 1 � − 1 | y − x | α , i ( x ) � ∀ i ∈ I , x , y ∈ V .

Introduction WSC and topological pressures Main results Problems Open set condition Open set condition (OSC): ∃ bounded open ∅ � = U ⊂ X such that S i ( U ) ⊆ U ∀ i and S i ( U ) ∩ S j ( U ) = ∅ ∀ i � = j . Cone condition (CC) for E ⊂ R d : ∃ β, h > 0 s.t. ∀ x ∈ ∂ E , ∃ open cone C ( x , u x , β, h ) ⊂ E ◦ with vertex x , direction vector u x , central angle of Lebesgue measure β , and altitude h . Topological pressure: � 1 � R s P ( s ) = lim n ln i . n →∞ i ∈ I n

Introduction WSC and topological pressures Main results Problems Dimension result for IIFS under BDP and OSC Theorem (Mauldin-Urb´ anski, 1996) Assume BDP, OSC and CC, and let ξ := inf { t ≥ 0 : � P ( s ) < 0 } . Then dim H ( K ) = ξ. In particular, if � P ( ξ ) = 0 , then dim H ( K ) = ξ .

Introduction WSC and topological pressures Main results Problems Anomalous phenomena for IIFSs • M. Moran (1996): Even for similitudes satisfying OSC, it is possible to have H α ( K ) = 0 , where α = dim H ( K ) . (Nevertheless, for such IIFSs, H α ( K ) < ∞ . ) • Mauldin-Urb´ anski (1996): Under BDP and OSC, its possible to have dim H ( K ) < dim B ( K ) ≤ dim P ( K ) . • Szarek-Wedrychowicz (2004): OSC �⇒ SOSC. • Topological pressure functions need not have a zero. In fact, domain of various topological pressures could be empty.

Introduction WSC and topological pressures Main results Problems Weak separation condition for IIFSs For 0 < b < 1, let I b = { i = ( i 1 , . . . , i n ) : R i ≤ b < R i 1 ··· i n − 1 } and A b = { S i : i ∈ I b } . Definition (a) Weak separation condition (WSC): ∃ invariant subset D ⊆ X with D ◦ � = ∅ , called a WSC set , and a constant γ ∈ N such that � � sup # τ ∈ A b : x ∈ τ ( D ) ≤ γ for all b ∈ (0 , 1) . (2.1) x ∈ X (b) If E ⊆ X is an invariant set and (2.1) holds with E replacing D, we call E a pre-WSC set . Thus, any pre-WSC set that has a nonempty interior is a WSC set.

Introduction WSC and topological pressures Main results Problems Example for WSC Example √ Let X = [0 , 1] , 0 < r < (2 − 2) / 2 ≈ 0 . 292893 . . . , r (2 − r ) / (1 − r ) < t < 1 − r, and S 2 k ( x ) = r k x + t (1 − r k − 1 ) , S 1 ( x ) = rx + (1 − r ) , S 2 k +1 ( x ) = r k x + t (1 − r k − 1 ) + r k (1 − r ) , k ≥ 1 . Then the IIFS does not satisfy OSC, but BDP holds and WSC holds with D = X.

Introduction WSC and topological pressures Main results Problems Figure for the example X=[0,1] n=1 n=2 K 0 0.2 0.4 0.6 0.8 1 Figure: First two iterations of the set X = [0 , 1] under the IIFS, with r = 1 / 5 and t = 1 / 2. The limit set K is also shown.

Introduction WSC and topological pressures Main results Problems Topological pressure Let S n = S n ( I ) := { S i : i ∈ I n } . Definition Upper and lower topological pressure functions: � � 1 1 R s R s P ( s ) := lim n ln φ , P ( s ) := lim n ln φ . n →∞ n →∞ φ ∈S n φ ∈S n If P ( s ) = P ( s ) , we denote the common value by P ( s ) and call P the topological pressure function . Define dom P = { s ∈ R : P ( s ) < ∞} ( Domain of P ) .

Introduction WSC and topological pressures Main results Problems Topological pressure properties • BDP ⇒ P V , P V are independent of V . • Assume BDP and WSC. Then [ d , ∞ ) ⊆ dom P , the limit defining P exists, P is strictly decreasing, convex on dom P and continuous on ( dom P ) ◦ .

Introduction WSC and topological pressures Main results Problems Dimension result for FIFS under BDP and WSC Theorem (Lau-X.Wang-N., 2009) Assume that a FIFS satisfies BDP and WSC. Then (a) α := dim H ( F ) = dim P ( F ) = dim B ( F ) ; 0 < H α ( F ) ≤ P α ( F ) < ∞ . (b)

Introduction WSC and topological pressures Main results Problems Dimension formula Theorem (Q. Deng-N., 2011) Assume that a FIFS satisfies BDP and WSC. Then dim H ( K ) is the unique zero of P. This result extends those by Y.Wang-N., 2001 and Lau-N. 2007 for similitudes satisfying FTC.

Introduction WSC and topological pressures Main results Problems Finite weak separation condition Another natural extension of WSC to IIFSs. Let F = F ( I ) := { J ⊂ I : J is finite } be the collection of all finite subsets of I . Definition Finite weak separation condition (FWSC): ∀ J ∈ F ( I ) , the FIFS { S j } j ∈ J satisfies WSC.

Introduction WSC and topological pressures Main results Problems FWSC is strictly weaker than WSC IIFS satisfying FWSC but not WSC. Example Let X = [0 , 1] and S k , i := x 2 k + i i = 0 , 1 , . . . , 2 k − 1 , 2 k , k ∈ N . That is, for each k, S k , i [0 , 1] , i = 0 , 1 , . . . , 2 k − 1 , is the union of all nonoverlapping dyadic intervals in [0 , 1] with length 1 / 2 k . Then K = [0 , 1] and the IIFS satisfies FWSC but not WSC.

Introduction WSC and topological pressures Main results Problems Topological pressure star Definition For each J ∈ F , let P J be the topological pressure function for the FIFS { S i } i ∈ J , i.e., � 1 R s P J ( s ) = lim n ln σ . n →∞ σ ∈S n ( J ) Define P ∗ ( s ) := sup P J ( s ) . J ∈F

Introduction WSC and topological pressures Main results Problems Auxiliary topological pressure Definition For any b ∈ (0 , 1) , define � � 1 1 R s R s Q ( s ) := lim − ln b ln τ , Q ( s ) := lim − ln b ln τ , b → 0 + b → 0 + τ ∈A b τ ∈A b and let Q ( s ) denote the common value if Q ( s ) = Q ( s ) .

Introduction WSC and topological pressures Main results Problems “Zeros” of topological pressures For each J ∈ F , denote the limit set of the FIFS { S i } i ∈ J by K J . Define α J := dim H ( K J ) , α := sup { α J : J ∈ F} , ˆ ξ ∗ := inf { s ≥ 0 : P ∗ ( s ) < 0 } , ξ := inf { s ≥ 0 : P ( s ) < 0 } , ζ := inf { s ≥ 0 : Q ( s ) < 0 } , ζ := inf { s ≥ 0 : Q ( s ) < 0 } .

Introduction WSC and topological pressures Main results Problems Main results Theorem (N-Tong) Assume BDP and WSC. (a) If K is a pre-WSC set, then α = ξ ∗ ≤ ξ. dim H ( K ) = ζ = ζ = � (b) If a WSC set D satisfies CC, then D is a WSC set. In particular, K is a pre-WSC set and thus the conclusion of part (a) holds.

Introduction WSC and topological pressures Main results Problems Outline of Proof • Combining Lau-N-X. Wang (2009) and Q. Deng-N(2011), we have the following key lemma: Lemma Assume BDP and WSC hold and K is a pre-WSC set. Then for any J ∈ F and any b ∈ (0 , 1) , � R α J ≤ c α J 1 γ. τ τ ∈A b • This lemma allows us to obtain the lower bound: ζ ≤ ζ ≤ dim H ( K ). • The upper bound can be obtained more easily by using covers provided by the definition of various topological pressures.