Dimension of stationary measures with infinite entropy Adam - PowerPoint PPT Presentation

Dimension of stationary measures with infinite entropy Adam ´ Spiewak University of Warsaw Student/PhD Dynamical Systems seminar June 05, 2020 Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Based on a preprint ”Dimension of Gibbs measures with infinite entropy” by Felipe P´ erez. Introduction Consider a contractive IFS f 1 , ..., f k : [0 , 1] → [0 , 1] and the corresponding coding map ∞ π : { 1 , ..., k } N → [0 , 1] , π ( a 1 , a 2 , ... ) = � f a 1 ◦ . . . ◦ f a n ([0 , 1]) . n =1 Let ν be a shift-invariant and ergodic measure on { 1 , . . . , k } N . We are interested in geometric properties of the measure µ = π ∗ ν on [0 , 1]. If ν = ( p 1 , ..., p k ) ⊗ N , then measure µ is the stationary measure for the random system ( { f 1 , ..., f k } , ( p 1 , ..., p k )), i.e. it satisfies k � µ = p j ( f j ) ∗ µ. j =1 Adam ´ Spiewak Dimension of stationary measures with infinite entropy

f 1 ( x ) = 1 3 x , f 2 ( x ) = 1 3 x + 2 3 , ν = ( 1 3 , 2 3 ) ⊗ N 1 0 1 f 1 f 2 1 2 3 3 0 1 2 1 3 3 f 1 ◦ f 1 f 1 ◦ f 2 f 2 ◦ f 1 f 2 ◦ f 2 1 2 2 4 9 9 9 9 0 1 2 1 2 7 8 1 9 9 3 3 9 9 1 2 2 4 2 4 4 8 27 27 27 27 27 27 27 27 π (1 , 2 , 1 , . . . ) π (2 , 1 , 1 , . . . ) Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Local dimensions Definition Let µ be a Borel probability measure on R n . Define lower and upper local dimension of µ at point x ∈ supp ( µ ) as log µ ( B ( x , r )) log µ ( B ( x , r )) d ( µ, x ) = lim inf and d ( µ, x ) = lim sup . log r log r r → 0 r → 0 If the limit exists, then µ ( B ( x , r )) ∼ r d ( µ, x ) . µ is called exact dimensional if d ( µ, x ) = d ( µ, x ) = const almost surely. Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Definition Lower and upper Hausdorff dimensions of µ : dim H ( µ ) = ess inf d ( µ, x ) , dim H ( µ ) = ess sup d ( µ, x ) x ∼ µ x ∼ µ Lower and upper packing dimensions of µ : dim P ( µ ) = ess inf d ( µ, x ) , dim P ( µ ) = ess sup d ( µ, x ) x ∼ µ x ∼ µ For exact dimensional measures, all of the above coincide. Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Proposition Let F = { f 1 , ..., f k } be a contractive IFS on [0 , 1] consisting of similiarities, i.e. f i ( x ) = r i x + t i , r i ∈ (0 , 1) . Assume that sets f i ([0 , 1]) , i = 1 , ..., k have disjoint interiors. Let p = ( p 1 , ..., p k ) be a probability vector and let µ be the stationary measure µ = π ∗ ( p ⊗ N ). Then µ is exact dimensional with k p i log 1 � p i Lyapunov exponent = h ( µ ) entropy i =1 d ( µ, x ) = λ ( µ ) := k − � p i log r i i =1 almost surely. This formula holds also for (well-behaved) infinite IFS and general ergodic measures, as long as h ( µ ) and λ ( µ ) are finite. Main question : what if h ( µ ) and λ ( µ ) are infinite? Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Proof: For a = ( a 1 , a 2 , . . . ) ∈ { 1 , . . . , k } N and n ∈ N define the n-th level cylinder I ( a 1 , . . . , a n ) = f a 1 ◦ . . . ◦ f a n ([0 , 1]). For x = π ( a ), let I n ( x ) be the n-th level cylinder containing x (it is unique for µ -almost every x ), hence if x = π ( a 1 , a 2 , . . . ) then I n ( x ) = I ( a 1 , . . . , a n ). 0 1 f a 1 ◦ . . . f a n π ( a ) = x 0 1 I n ( x ) = I ( a 1 , ..., a n ) log µ ( B ( π ( a ) , r )) We want to calculate lim for almost every a . First we will log r r → 0 calculate the symbolic dimension log µ ( I n ( x )) δ ( x ) := lim log | I n ( x ) | n →∞ Adam ´ Spiewak Dimension of stationary measures with infinite entropy

For x = π ( a 1 , a 2 , . . . ) log µ ( I n ( x )) log µ ( I ( a 1 , . . . , , a n )) log p a 1 · · · p a n δ ( x ) = lim = lim = lim = log | I n ( x ) | log | I ( a 1 , . . . , a n ) | log r a 1 · · · r a n n →∞ n →∞ n →∞ n k 1 � log p a j � p i log p i n = h ( µ ) j =1 i =1 = lim = λ ( µ ) ν -a.s. n n →∞ k 1 � log r a j � p i log r i n j =1 i =1 How to relate δ ( x ) with d ( x ) and d ( x )? Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Fix x = π ( a ) and r > 0. There exists unique n = n ( r ) ∈ N such that | I n ( x ) | < r ≤ | I n − 1 ( x ) | . Note that n ( r ) → ∞ as r → 0 I n − 1 ( x ) f a n I n ( x ) x B ( x , r ) log µ ( B ( x , r )) ≥ log µ ( I n ( x )) and log r ≤ log | I n − 1 ( x ) | , hence log µ ( B ( x , r )) ≤ log µ ( I n ( x )) log | I n − 1 ( x ) | = log µ ( I n ( x )) log | I n ( x ) | log | I n ( x ) | · log | I n − 1 ( x ) | → δ ( x ) as log r min { r i }| I n − 1 ( x ) | ≤ | I n ( x ) | ≤ max { r i }| I n − 1 ( x ) | . Adam ´ Spiewak Dimension of stationary measures with infinite entropy

We have proven d ( x ) ≤ δ ( x ) almost surely. One can similarly prove d ( x ) ≥ δ ( x ). Gauss system - basic example of an infinite IFS Gauss map T : (0 , 1] → (0 , 1] , T ( x ) = 1 x − ⌊ 1 x ⌋ Gauss system 1 f i : [0 , 1] → [0 , 1] , f i ( x ) = x + i , x + i } ∞ 1 F = { f i ( x ) = i =1 1 π : N N → [0 , 1] , π ( a ) = 1 a 1 + 1 a 2 + a 3 + 1 ... Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Setting Assumptions Let T : (0 , 1] → (0 , 1] be such that there exists a decomposition ∞ (0 , 1] = � I ( n ) into closed intervals with disjoint interiors with lengths n =1 r n = | I ( n ) | such that ∞ (1) T is C 2 on � Int ( I ( n )) n =1 ∞ � � | ( T k ) ′ ( x ) | : x ∈ � (2) there exists k ≥ 1 such that inf Int ( I ( n )) > 1 n =1 | T ′′ ( x ) | (3) sup sup | T ′ ( y ) || T ′ ( z ) | < ∞ (R´ enyi’s condition) n ∈ N x , y , z ∈ I ( n ) (4) T ( I ( n )) = (0 , 1] , I ( n + 1) < I ( n ) and r n +1 < r n (5) 0 < K ≤ r n +1 / r n ≤ K ′ < ∞ for some constants K , K ′ n →∞ n t r n < ∞} satisfies (6) r n decays polynomially, i.e. α = sup { t ≥ 0 : lim 1 < α < ∞ (7) T is orientation preserving on each I ( n ) Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Adam ´ Spiewak Dimension of stationary measures with infinite entropy

For a 1 , . . . , a n ∈ N define the n -th level cylinder I ( a 1 , ..., a n ) = I ( a 1 ) ∩ T − 1 ( I ( a 2 )) ∩ . . . ∩ T − ( n − 1) ( I ( a n )) Let Σ = N N and define the natural projection π : Σ → (0 , 1] by ∞ � π ( a 1 , a 2 , . . . ) = I ( a 1 , . . . , a n ) . n =1 π is a continuous bijection satisfying π ◦ σ = T ◦ π , where σ is the left shift on Σ. For a symbolic cylinder C ( a 1 , ..., a n ) ⊂ Σ we have π ( C ( a 1 , ..., a n )) = I ( a 1 , ..., a n ) . ∞ ∞ T − n ( ∂ I ( k )). For every x ∈ (0 , 1] \ O there is a unique Let O = � � n =0 k =1 n -th level cylinder I n ( x ) containing x . Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Proposition (consequence of the R´ enyi condition) There exists D ≥ 1 such that 0 < D − 1 ≤ | ( T n ) ′ ( x ) | · | I ( a 1 , . . . , a n ) | ≤ D holds for every sequence ( a 1 , . . . , a n ) ∈ N n and every x ∈ Int ( I ( a 1 , . . . , a n )) Proposition (consequence of the previous one) There exist D 1 , D 2 > 0 such that for every ( a 1 , . . . , a n ) ∈ N n and m ∈ N we have n (1) | log | I ( a 1 , . . . , a n ) | − � log r a k | ≤ nD 1 + D 2 k =1 n − 1 m m (2) | log | � I ( a 1 , . . . , a n + j ) | − � log r a k − log( � r a n + j ) | ≤ nD 1 + D 2 j =0 k =1 j =0 ∞ n − 1 ∞ � � � (3) | log | I ( a 1 , . . . , a n + j ) | − log r a k − log( r a n + j ) | ≤ nD 1 + D 2 j =0 k =1 j =0 Adam ´ Spiewak Dimension of stationary measures with infinite entropy

n � Proof of | log | I ( a 1 , . . . , a n ) | − log r a k | ≤ nD 1 + D 2 k =1 By R´ enyi’s condition − log D ≤ log | I ( a 1 , . . . , a n ) | + log( T n ) ′ ( x ) ≤ log D for x ∈ I ( a 1 , . . . , a n ) . We have n − 1 � log( T n ) ′ ( x ) = log T ′ ( T k x ) k =0 and, as T k x ∈ I ( a k +1 ), − log D ≤ log | I ( a k +1 ) | + log T ′ ( T k x ) = log r a k +1 + log T ′ ( T k ) ≤ log D , hence summing over k = 0 , ..., n − 1 n − 1 n � � log T ′ ( T k x ) + − n log D ≤ log r a k ≤ n log D . k =0 k =1 � Adam ´ Spiewak Dimension of stationary measures with infinite entropy

Gibbs measures Definition An ergodic shift invariant measure ν on Σ is called a Gibbs measure associated to the potential ϕ : Σ → R if there exist constants P ∈ R and A , B > 0 such that for every point x ∈ C ( a 1 , . . . a n ) ν ( C ( a 1 , . . . , a n )) A ≤ � ≤ B , � exp − nP + S n ϕ ( x ) n − 1 ϕ ( σ k x ) is the Birkhoff sum of ϕ at x . We will assume where S n ϕ = � k =0 P = 0 (otherwise take ϕ − P as the potential). Examples (1) Bernoulli measures (2) Markov measures (3) ( π − 1 ) ∗ Leb , where π is the natural projection for the Gauss map Adam ´ Spiewak Dimension of stationary measures with infinite entropy