Dimensions of invariant measures for affine iterated function - - PowerPoint PPT Presentation

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

Consider an affine IFS {Si}ℓ

i=1 on Rd of the form

Si(x) = Aix + ai, i = 1, . . . , ℓ, where Ai are invertible d × d matrices with Ai < 1 and ai ∈ Rd.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Affine IFSs and self-affine sets

Consider an affine IFS {Si}ℓ

i=1 on Rd of the form

Si(x) = Aix + ai, i = 1, . . . , ℓ, where Ai are invertible d × d matrices with Ai < 1 and ai ∈ Rd. The attractor K of this IFS, is the unique nonempty compact set satisfying K =

ℓ

• i=1

Si(K). K is called the self-affine set generated by {Si}ℓ

i=1. If Si 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 {Si}ℓ

i=1 be an affine IFS on Rd. Let Σ = {1, . . . , ℓ}Z.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Symbolic dynamics and the coding map

Let {Si}ℓ

i=1 be an affine IFS on Rd. Let Σ = {1, . . . , ℓ}Z.

Define the coding map π : Σ → Rd, by π(x) = lim

n→∞ Sx0 ◦ Sx1 ◦ · · · ◦ Sxn(0),

x = (xi)∞

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. When m = ∞

i=−∞{p1, . . . , pℓ} on Σ, µ = m ◦ π−1 is a

self-affine measure which satisfies µ =

ℓ

• i=1

pi µ ◦ S−1

i

.

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 dimloc(µ, x) = lim sup

r→0

log µ(B(x, r)) log r , dimloc(µ, x) = lim inf

r→0

log µ(B(x, r)) log r , If dimloc(µ, x) = dimloc(µ, x), the common value is denoted as dimloc(µ, x) and is called the local dimension of µ at x. Moreover, µ is said to be exact dimensional if dimloc(µ, x) = C for µ-a.e. x ∈ Rd.

De-Jun Feng Dimensions of invariant measures for affine IFSs

(Young, 1982) If µ is exact dimensional, then dimHµ = dimHµ = C, where dimHµ = inf{dimH F : µ(F) > 0 and F is a Borel set}, dimHµ = inf{dimH F : µ(Rd \ 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 δu, δs, 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 δu, δs, the local dimensions along stable and unstable local manifolds. (Barreira-Pesin-Schmeling, 1999): in high-dimensional C 1+δ diffeomorphism case, local dimension = δu + δs; 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 δu, δs, the local dimensions along stable and unstable local manifolds. (Barreira-Pesin-Schmeling, 1999): in high-dimensional C 1+δ diffeomorphism case, local dimension = δu + δs; 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 δu, δs, the local dimensions along stable and unstable local manifolds. (Barreira-Pesin-Schmeling, 1999): in high-dimensional C 1+δ diffeomorphism case, local dimension = δu + δs; 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 Ai commute. ( with a Ledarppier-Young type dimension formula)

De-Jun Feng Dimensions of invariant measures for affine IFSs

(Barany-K¨ aenm¨ aki, 2015): Exact dimensionality of

self-affine measures in R2 . self-affine measures in Rn with n distinct Lyapunov exponents. quasi-self-affine measures in Rn (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 → Rd be the coding map associated with an affine IFS {Si}ℓ

i=1.

Theorem (F.) Let m be a shift-invariant measure on Σ and µ = m ◦ π−1. Then dimloc(µ, z) exists for µ-a.e. z. If m is ergodic, then µ is exact dimensional. Furthermore, dimH 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, dimH m ◦ π−1 =

s−1

• i=0

hi − hi+1 λi+1 = h0 λ1 −

s−1

• i=1

hi 1 λi − 1 λi+1

• − hs

λs , where hi = Hm(P|ξi), P = {[i]0 : i = 1, 2, . . . , ℓ}. Roughly speaking, dimH m ◦ π−1 =

s

• i=1

fibre entropies Lyapunov exponents.

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(Rd) defined by x = (xi)+∞

i=−∞ → A−1 x0 .

De-Jun Feng Dimensions of invariant measures for affine IFSs

Theorem (Oseledets 1968) ∃ s, k1, . . . , ks with k1 + · · · + ks(x) = d, and real numbers λ1 < · · · < λs so that for m-a.e x = (xi)+∞

i=−∞,

Rd = E 1

x ⊕ · · · ⊕ E s x

such that the following properties hold: (i) dim E i

x = ki and A−1 x0 E i x = E i σx for 1 ≤ i ≤ s.

(ii) For 1 ≤ i ≤ s and v ∈ E i

x\{0},

lim

n→+∞

1 n log A−1

xn . . . A−1 x0 v

= λi lim

n→+∞

1 n log Ax−n . . . Ax−1v = −λi

De-Jun Feng Dimensions of invariant measures for affine IFSs

Filtrations

For m-a.e. x, set V i

x = ⊕s j=i+1E j x,

i = 0, 1, . . . , s − 1, and V s

x = {0}.

Then Rd = V 0

x ⊃ V 1 x ⊃ · · · ⊃ V s x = {0}.

These linear spaces V i

x only depend on i and x− := (xj)−1 j=−∞.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Construct a family of measurable partitions ξ0, . . . , ξs of Σ as follows: ξi(x) := {y ∈ Σ : y− = x−, πy − πx ∈ V i

x},

here ξi(x) denotes the ξi-atom that contains x.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Application 1: Dimension of self-affine sets

Let m be an ergodic measure on Σ. Write π = πA,a, where A = (A1, . . . , Aℓ) and a = (a1, . . . aℓ). Fact: a → hi(m, A, a) are upper semi-continuous (as a variant of Rapaport 2015) Theorem (F.) The mapping a → dimH m ◦ (πA,a)−1 is lower semi-continuous.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Let KA,a be the self-affine set generated by {Si = Aix + ai}ℓ

i=1.

Corollary (F.) Assume that Ai < 1/2. Then {a ∈ Rℓd : dimH KA,a = min{d, dimAffine A}} is of first Baire category in Rℓd. Remark: By Falconer (1988)-Solomyak (1998), the above set is of zero Lebesgue measure.

De-Jun Feng Dimensions of invariant measures for affine IFSs

How to find concrete cases of (A, a) so that dimH KA,a = min{d, dimAffine A}? Recent advances by Barany 2015, Falconer-Kempton 2015, and Barany-K¨ aenm¨ aki 2015, Barany-Rams 2015, Rapaport 2015, Morris-Shmerkin 2016. Use Exact dimensionality+ Marstand projection theorem

De-Jun Feng Dimensions of invariant measures for affine IFSs

Set τ : Σ → G(d, ks) by x → V s−1

x

= E s

x . ( τ is well defined for

m-a.e. x, and only depends on A and m). Theorem Assume that m is a Bernoulli product measure or Gibbs measure. Under the Strong Separation Condition, if dimHm ◦ τ −1 + dimLY m > (d − ks)(ks + 1). Then dimH m ◦ π−1 = dimLY m. This is the restatement of Rapaport (2015), but with less

• assumptions. Similarly we can sharpen the result of

Barany-K¨ aenm¨ aki (2015).

De-Jun Feng Dimensions of invariant measures for affine IFSs

Recall that dimLY m = t where t is the unique value so that hm(σ) + φt

∗(m) = 0, here φt(·)

denotes the singular value function. Hence under certain condition on the linear parts A of the IFS, and SSC, one have dimH KA,a = min{d, dimAffine A}. Related results: Bourgain (2012), Barany-Rams (2015), Hochman-Solomyak (2016), Morris-Shmerkin (2016)

De-Jun Feng Dimensions of invariant measures for affine IFSs

Application 2: Dimension conservation

Let η be a probability measure on Rn and W is a subspace of Rn. Definition (Furstenberg) Say that η satisfies the dimension conservation along direction W if dimH η = dimH ηW ,x + dimH(PW ⊥)∗η for η-a.e. x, where PW ⊥ denote the orthogonal projection onto W ⊥, and ηW ,x the slice measures of µ along W .

De-Jun Feng Dimensions of invariant measures for affine IFSs

Theorem (F.) Let µ = m ◦ π−1 be a self-affine measure or the push-forward of a Gibbs measure. Then η satisfies the dimension conservation along the direction W = V i

x for m-a.e x, i = 1, . . . , s − 1.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Generalizations: stationary measures for average contractive affine IFSs

Consider a finite IFS {Aix + ai}i∈Λ, where Ai can be non-contractive and non-invertible. Suppose that m is an ergodic measure on Σ := ΛZ such that the IFS is average contractive in the sense that λ := limn→∞ 1

n

• log Ax1 . . . Axndm(x) < 0.

Then our main results (e.g. exact dimensionality + dim formula) remain valid for m ◦ π−1.

De-Jun Feng Dimensions of invariant measures for affine IFSs

Thank you !!!

De-Jun Feng Dimensions of invariant measures for affine IFSs