Dimensions of invariant measures for affine iterated function systems De-Jun Feng The Chinese University of Hong Kong Workshop on GMT, University of Warwick July 13, 2017 De-Jun Feng Dimensions of invariant measures for affine IFSs
Affine IFSs and self-affine sets De-Jun Feng Dimensions of invariant measures for affine IFSs
Affine IFSs and self-affine sets i =1 on R d of the form Consider an affine IFS { S i } ℓ S i ( x ) = A i x + a i , i = 1 , . . . , ℓ, where A i are invertible d × d matrices with � A i � < 1 and a i ∈ R d . De-Jun Feng Dimensions of invariant measures for affine IFSs
Affine IFSs and self-affine sets i =1 on R d of the form Consider an affine IFS { S i } ℓ S i ( x ) = A i x + a i , i = 1 , . . . , ℓ, where A i are invertible d × d matrices with � A i � < 1 and a i ∈ R d . The attractor K of this IFS, is the unique nonempty compact set satisfying ℓ � K = S i ( K ) . i =1 K is called the self-affine set generated by { S i } ℓ i =1 . If S i are similitudes, K is called self-similar. De-Jun Feng Dimensions of invariant measures for affine IFSs
A self-affine set De-Jun Feng Dimensions of invariant measures for affine IFSs
A self-affine set De-Jun Feng Dimensions of invariant measures for affine IFSs
A self-affine set De-Jun Feng Dimensions of invariant measures for affine IFSs
A self-affine set De-Jun Feng Dimensions of invariant measures for affine IFSs
A fundamental question in fractal geometry and dynamical systems Q: How to calculate the dimensions (Hausdorff & box-counting) of self-affine sets? De-Jun Feng Dimensions of invariant measures for affine IFSs
A fundamental question in fractal geometry and dynamical systems Q: How to calculate the dimensions (Hausdorff & box-counting) of self-affine sets? Partial results: McMullen (1984), Bedford (1984), Falconer (1988), Heuter-Lalley (1992), Kenyon-Peres (1996), Solomyak (1998), Hochman (2014), Barany (2015), Falconer-Kempton (2015), Barany-K¨ aenm¨ aki (2015), Hochman-Solomyak (2016), Morris-Shmerkin (2016), Das-Simmons (2016), ..., etc. De-Jun Feng Dimensions of invariant measures for affine IFSs
A fundamental question in fractal geometry and dynamical systems Q: How to calculate the dimensions (Hausdorff & box-counting) of self-affine sets? Partial results: McMullen (1984), Bedford (1984), Falconer (1988), Heuter-Lalley (1992), Kenyon-Peres (1996), Solomyak (1998), Hochman (2014), Barany (2015), Falconer-Kempton (2015), Barany-K¨ aenm¨ aki (2015), Hochman-Solomyak (2016), Morris-Shmerkin (2016), Das-Simmons (2016), ..., etc. A key issue is to estimate the dimension of “good” measures on self-affine sets. De-Jun Feng Dimensions of invariant measures for affine IFSs
Symbolic dynamics and the coding map Let { S i } ℓ i =1 be an affine IFS on R d . Let Σ = { 1 , . . . , ℓ } Z . De-Jun Feng Dimensions of invariant measures for affine IFSs
Symbolic dynamics and the coding map Let { S i } ℓ i =1 be an affine IFS on R d . Let Σ = { 1 , . . . , ℓ } Z . Define the coding map π : Σ → R d , by x = ( x i ) ∞ π ( x ) = lim n →∞ S x 0 ◦ S x 1 ◦ · · · ◦ S x n (0) , i = −∞ . Then π (Σ) = K . De-Jun Feng Dimensions of invariant measures for affine IFSs
Stationary (invariant) measures for IFS. Let m be a shift-invariant measure on Σ. Definition The push-forward µ = m ◦ π − 1 on K is called a stationary (or invariant) measure for the IFS. If m is ergodic, then µ is called ergodic stationary. i = −∞ { p 1 , . . . , p ℓ } on Σ , µ = m ◦ π − 1 is a When m = � ∞ self-affine measure which satisfies ℓ � p i µ ◦ S − 1 µ = . i i =1 De-Jun Feng Dimensions of invariant measures for affine IFSs
Our Targets Dimensional properties of stationary measures for affine IFSs. Dimensions of self-affine sets De-Jun Feng Dimensions of invariant measures for affine IFSs
Notation: local dimensions and exact-dimensionality The local upper and lower dimensions of a prob. measure µ at x log µ ( B ( x , r )) dim loc ( µ, x ) = lim sup , log r r → 0 log µ ( B ( x , r )) dim loc ( µ, x ) = lim inf , log r r → 0 If dim loc ( µ, x ) = dim loc ( µ, x ), the common value is denoted as dim loc ( µ, x ) and is called the local dimension of µ at x . Moreover, µ is said to be exact dimensional if dim loc ( µ, x ) = C for µ -a.e. x ∈ R d . De-Jun Feng Dimensions of invariant measures for affine IFSs
(Young, 1982) If µ is exact dimensional, then dim H µ = dim H µ = C , where dim H µ = inf { dim H F : µ ( F ) > 0 and F is a Borel set } , dim H µ = inf { dim H F : µ ( R d \ F ) = 0 and F is a Borel set } . De-Jun Feng Dimensions of invariant measures for affine IFSs
A natural question Q : Is every ergodic stationary measure for an affine IFS exact dimensional? De-Jun Feng Dimensions of invariant measures for affine IFSs
Historical remarks: (1) smooth cases There is a long history for the problem of existence of local dimensions and exact dimensionality: ( Bowen 1979 ): Ergodic invariant measures on C 1+ δ -conformal repellers. ( Young, 1982 ): Any ergodic hyperbolic measure invariant under a C 1+ δ (compact) surface diffeomorphism is always exactly dimensional. De-Jun Feng Dimensions of invariant measures for affine IFSs
( Ledrappier-Young, 1985 ): in high-dimensional C 2 diffeomorphism case, the existence of δ u , δ s , the local dimensions along stable and unstable local manifolds. De-Jun Feng Dimensions of invariant measures for affine IFSs
( Ledrappier-Young, 1985 ): in high-dimensional C 2 diffeomorphism case, the existence of δ u , δ s , the local dimensions along stable and unstable local manifolds. ( Barreira-Pesin-Schmeling, 1999 ): in high-dimensional C 1+ δ diffeomorphism case, local dimension = δ u + δ s ; an answer to Eckmann-Ruelle conjecture. De-Jun Feng Dimensions of invariant measures for affine IFSs
( Ledrappier-Young, 1985 ): in high-dimensional C 2 diffeomorphism case, the existence of δ u , δ s , the local dimensions along stable and unstable local manifolds. ( Barreira-Pesin-Schmeling, 1999 ): in high-dimensional C 1+ δ diffeomorphism case, local dimension = δ u + δ s ; an answer to Eckmann-Ruelle conjecture. ( Qian-Xie, 2008 ): C 2 expanding endomorphisms. De-Jun Feng Dimensions of invariant measures for affine IFSs
( Ledrappier-Young, 1985 ): in high-dimensional C 2 diffeomorphism case, the existence of δ u , δ s , the local dimensions along stable and unstable local manifolds. ( Barreira-Pesin-Schmeling, 1999 ): in high-dimensional C 1+ δ diffeomorphism case, local dimension = δ u + δ s ; an answer to Eckmann-Ruelle conjecture. ( Qian-Xie, 2008 ): C 2 expanding endomorphisms. ( Shu, 2010 ): C 1+ δ non-degenerate hyperbolic endomorphisms. De-Jun Feng Dimensions of invariant measures for affine IFSs
Historical remarks: (2) self-affine cases Some very special examples of self-affine measures ( Bedford 84, McMullen 84 , Lalley-Gatzouras 92 , Kenyon-Peres 96 , Baranski 07 , etc). With precise dimension formulas. “Typical” self-affine / ergodic stationary measures ( K¨ aenm¨ aki 2004, Jordan-Pollicott-Simon 2007, Jordan, Rossi ) De-Jun Feng Dimensions of invariant measures for affine IFSs
( F. & Hu, 2009 ): Exact dimensionality for self-similar measures, self-conformal measures, more generally, ergodic stationary measures for conformal IFSs. ( with dimension = projection entropy / Lyapunov exponent . ) ergodic stationary measures (including self-affine measures) for those affine IFSs so that the linear parts A i commute. ( with a Ledarppier-Young type dimension formula) De-Jun Feng Dimensions of invariant measures for affine IFSs
( Barany-K¨ aki, 2015 ): Exact dimensionality of aenm¨ self-affine measures in R 2 . self-affine measures in R n with n distinct Lyapunov exponents. quasi-self-affine measures in R n ( n ≥ 3) under a technical assumption (“dominated splittings”). Remark : B-K improves previous results of Falconer-Kempton 2015, Barany 2015 . A related result by Rapaport 2015 . De-Jun Feng Dimensions of invariant measures for affine IFSs
Our main result Let π : Σ = { 1 , . . . , ℓ } Z → R d be the coding map associated with an affine IFS { S i } ℓ i =1 . Theorem (F.) Let m be a shift-invariant measure on Σ and µ = m ◦ π − 1 . Then dim loc ( µ, z ) exists for µ -a.e. z. If m is ergodic, then µ is exact dimensional. Furthermore, dim H µ satisfies a Ledrappier-Young type formula. De-Jun Feng Dimensions of invariant measures for affine IFSs
Theorem (F.) For any ergodic measure m on Σ , s − 1 h i − h i +1 � dim H m ◦ π − 1 = λ i +1 i =0 � 1 s − 1 � h 0 1 − h s � = − h i − , λ 1 λ i λ i +1 λ s i =1 where h i = H m ( P| ξ i ) , P = { [ i ] 0 : i = 1 , 2 , . . . , ℓ } . Roughly speaking, s fibre entropies dim H m ◦ π − 1 = � Lyapunov exponents . i =1 De-Jun Feng Dimensions of invariant measures for affine IFSs
Dimension formula of ergodic stationary measures for affine IFSs Let m be an ergodic measure on Σ. Consider the local constant matrix cocycle M : Σ → GL ( R d ) defined by x = ( x i ) + ∞ i = −∞ �→ A − 1 x 0 . De-Jun Feng Dimensions of invariant measures for affine IFSs
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