On the dimension of diagonally a ffi ne self-a ffi ne sets and - - PowerPoint PPT Presentation

On the dimension of diagonally affine self-affine sets and overlaps.

K´ aroly Simon

Department of Stochastics Institute of Mathematics Budapest University of Technolugy and Economics www.math.bme.hu/˜simonk Joint work with Bal´ azs B´ ar´ any and Michal Rams

ICERM February 2016

• B. B´

ar´ any, M. Rams, K. Simon Diagonally Self-affine sets and measures February 19, 2016 1 / 43

Co-authors

B´ ar´ any Bal´ azs Univ. of Warwick/ TU Budapest (left) Michal Rams, Warsaw IMPAN (right)

• B. B´

ar´ any, M. Rams, K. Simon Diagonally Self-affine sets and measures February 19, 2016 2 / 43

Introduction

1

Introduction Abstract Notation An old result of B´ ar´ any Bal´ azs Hochman-condition

2

Results Theorem A Theorem B Proposition C Theorem D

3

Feng-Hu Theorem about Ledrappier-Young formula Notation The self-similar case Diagonally self-affine case

4

Stuff what I am going to have no time for

• B. B´

ar´ any, M. Rams, K. Simon Diagonally Self-affine sets and measures February 19, 2016 3 / 43

Introduction Abstract

Abstract

In this talk we consider diagonally affine, planar IFS Φ = {Si(x, y) = (–ix + ti,1, —iy + ti,2)}m

i=1 .

Combining the techniques of Hochman and Feng-Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.

• B. B´

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Introduction Notation

Definitions

(1) Φ = {Si(x, y) = (–ix + ti,1, —iy + ti,2)}m

i=1 ,

where 0 < |–i|, |—i| < 1, and we assume that Si([0, 1]2) µ [0, 1]2. We call a Borel probability measure µ self-affine if it is compactly supported with support Λ and there exists a p = (p1, . . . , pm) probability vector such that (2) µ =

m ÿ i=1

pi µ ¶ S≠1

i

.

• B. B´

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Introduction Notation

Lyapunov exponent

The entropy and the Lyapunov exponents of µ: hµ := ≠

m ÿ i=1

pi log pi, and (3) ‰– := ≠

m ÿ i=1

pi log |–i|, ‰— := ≠

m ÿ i=1

pi log |—i|.

• B. B´

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Introduction Notation Λµ(t) t 1 2 3 4

− Λ

µ

( t ) hµ D(µ) slope= χ3 slope= χ2 slope= χ1 Lyapunov exponents: 0 < χ1 ≤ χ2 ≤ χ3 ≤ χ4

Figure: Definition of the Laypunov dimension D(µ)

• B. B´

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Introduction Notation

Lyapunov dimension by formula

If (4) k := max {i : 0 < h‹ ≠ ‰1(‹) ≠ · · · ≠ ‰i(‹)} Æ d ≠ 1, then we define the Lyapunov dimension of ‹: D(‹) := k + h‹ ≠ ‰1(‹) ≠ · · · ≠ ‰k(‹) ‰k+1(‹) ; If h‹ ≠ ‰1(‹) + · · · ≠ ‰d(‹) > 0 then we define D(‹) := d · h‹ ‰1(‹) + · · · + ‰d(‹), where h‹ is the entropy of the measure ‹.

• B. B´

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Introduction Notation

Lyapunov dimension on the plane

D(‹) :=

Y _ _ _ _ _ _ _ _ _ _ ] _ _ _ _ _ _ _ _ _ _ [

h‹ ‰1(‹), if h‹ Æ ‰1(‹); 1 + h‹ ≠ ‰1(‹) ‰2(‹) , if ‰1(‹) < h‹ Æ ‰1 + ‰2(‹); 2 · h‹ ‰1(‹) + ‰2(‹), if h‹ > ‰1(‹) + ‰2(‹). What does this mean exactly? See it in a special case:

• B. B´

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Introduction Notation

Lyapunov dimension in a very special case

Q f1(Q) λ λ λ f2(Q) f3(Q) ξ ξ ξ 1 1

Let ‹ be the self-affine measure which corresponds to p1 = p2 = p3 = 1

3.

‰1 = ≠ log ⁄ < ≠ log › = ‰2 Clearly, h‹ = log 3 D(‹) =

Y _ _ _ _ ] _ _ _ _ [

log 3 ≠ log ⁄, if ⁄ < 1

3;

log 3 ≠ log ⁄ ≠ log › , if ⁄ > 1

3;

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Introduction An old result of B´ ar´ any Bal´ azs

An old result

First we consider an old result due to Bal´ azs B´ ar´ any. Notation s– = dimB projxΛ, s— := dimB projyΛ. d– and d— are the solutions of the equations:

m ÿ i=1

–sα

i —dα≠sα i

= 1 and

m ÿ i=1

—sβ

i –dβ≠sβ i

= 1

• B. B´

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Introduction An old result of B´ ar´ any Bal´ azs

B´ ar´ any’s Theorem

Theorem 1.1 (B´ ar´ any (2011)) W.L.G. we may assume that Si([0, 1]2) µ [0, 1]2. Assume that {Si}m

i=1 satisfies Strong Separation Condition:

(5) Si([0, 1]2) fl Sj([0, 1]2) = ÿ. Then (6) dimB Λ = max {d–, d—} .

• B. B´

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Introduction Hochman-condition

Hochman-condition

We say that an IFS G = {fi(x)}iœS

• f similarities on the real line satisfies the

Hochman-condition if there exists an Á > 0 such that for every n > 0 min {∆(ı, ä) : ı, ä œ Sn, ı ”= ä} > Án, where ∆(ı, ä) =

Y ] [

Œ f Õ

ı (0) ”= f Õ ä (0)

|fı(0) ≠ fä(0)| f Õ

ı (0) = f Õ ä (0).

• B. B´

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Introduction Hochman-condition

Examples when Hochman Condition holds

If the parameters of the IFS G = {fi(x) = rix + ti}iœS

• f similarities are algebraic, i.e.

both ti and ri are algebraic numbers , then either the Hochman-condition holds or there is a complete overlap, that is, there exist n Ø 1, and ı ”= ä œ Sn such that fı(0) = fä(0) .

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Introduction Hochman-condition

Hochman Theorem

Suppose that an IFS Ψ = {rix + ti}m

i=1, |ri| < 1 of

contracting similarities on the real line satisfies the Hochman-condition. Let P := {p1, . . . , pm}N. Then for the measure µ = P ¶ Π≠1, dimH µ = min

I

1, hµ ‰

J

, where hµ is the entropy and ‰ is the Lyapunov exponent: hµ = ≠

M ÿ i=1

pi log pi and ‰ = ≠

ÿ pi log r1

.

• B. B´

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Introduction Hochman-condition

Hausdorff dimension of a measure

Here we recall the Hausdorff dimension of a probability measure µ, dimH µ = inf {dimH A : µ(A) = 1} = ess sup

µ≥x

lim inf

ræ0+

log µ(Br(x)) log r ,

• B. B´

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Introduction Hochman-condition

Families of self-similar IFS

Let I µ R be a compact parameter interval and m Ø 2. For every parameter t œ I given a self ≠ similar IFS on the line: Φt := {Ïi,t(x) = ri(t) · (x ≠ ai(t))}m

i=1 ,

where ri : I æ (≠1, 1) \ {0} and ai : I æ R are real analytic functions. Let Πt be the natural projection from Σ := {1, . . . , m}N to the attractor Λt of Φt.

• B. B´

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Introduction Hochman-condition

Families of self-similar IFS (cont.)

For every probability vector p := (p1, . . . , pm) the associated self-similar measure is ‹p,t := (Πt)ú(pN). Its similarity dimension is defined by dimS(‹p,t) :=

m q i=1 pi log pi m q i=1 pi log ri(t)

The similarity dimension

• f Λt is the solution s(t) of

r s(t)

1

(t) + · · · + r s(t)

m (t) = 1.

• B. B´

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Introduction Hochman-condition

Families of self-similar IFS (cont.)

We say that a parameter t œ I is exceptional if either dimH Λt < min {1, s(t)}

• r there exists a probability vector p := (p1, . . . , pm) such

that dimH(‹p,t) < min {1, dimS(‹p,t)} .

• B. B´

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Introduction Hochman-condition

Families of self-similar IFS (cont.)

Theorem 1.2 (Hochman) Assume that for i, j œ Σ = {1, . . . , m}N we have if Πt(i) = Πt(j) holds for all t œ I then i = j. Then both the Hausdorff and the packing dimension of the set of exceptional parameters are equal to 0.

• B. B´

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Results

1

Introduction Abstract Notation An old result of B´ ar´ any Bal´ azs Hochman-condition

2

Results Theorem A Theorem B Proposition C Theorem D

3

Feng-Hu Theorem about Ledrappier-Young formula Notation The self-similar case Diagonally self-affine case

4

Stuff what I am going to have no time for

• B. B´

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Results Theorem A

Theorem A

Theorem A Let Φ be an IFS of the form (1) and let µ be a self-affine measure of the form (2). Without loss of generality we may assume that ‰– Æ ‰— (i.e. the direction of y-axis is strong stable direction).

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Results Theorem A

Theorem A (cont.)

1

Suppose Φ– satisfies the Hochman-condition and hµ ‰– Æ 1. Then dimH µ = hµ ‰– .

2

Suppose Φ– and Φ— satisfy the Hochman-condition and hµ ‰— Æ 1 < hµ ‰– . Then dimH µ = 1 + hµ ≠ ‰– ‰— .

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Results Theorem B

Road towards Theorem B

As a consequence of Theorem A we can calculate the dimension of the attractor. Denote by s– and s— the similarity dimensions of the IFSs Φ– and Φ— respectively, i.e. s– and s— are the unique solutions of the equations (7)

m ÿ i=1

|–i|sα = 1, and

m ÿ i=1

|—i|sβ = 1.

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Results Theorem B

Theorem B

Theorem B Let Φ be an IFS of the form (1) and let Λ be the attractor of Φ. Without loss of generality we may assume that s— Æ s–.

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Results Theorem B

Theorem B (cont.)

1

Suppose Φ– satisfies the Hochman-condition and s– Æ 1. Then dimH Λ = dimB Λ = s–.

2

Suppose Φ– and Φ— satisfy the Hochman-condition and s— Æ 1 < s–. Then dimH Λ = dimB Λ = d , where d is the unique solution of

qm i=1 |–i||—i|d≠1 = 1 .

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Results Proposition C

Proposition C

Proposition C Let Φ be an IFS of the form (1). Let us assume that maxi”=j {|–i| + |–j|} < 1 and

m ÿ i=1

|—i| Æ 1. Then there exists a set T µ R2m such that dimP T Æ 2m ≠ 2 and for every (t1,1, . . . , tm,1, t1,2, . . . , tm,2) œ R2m \ T the statements of Theorem A and Theorem B hold.

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Results Proposition C

We obtained these estimates by using the method of Fraser and Shmerkin. Peres and Shmerkin showed that for every self-similar set in R or R2 for any Á > 0 there exists a self-similar set contained in the original one with dimension Á-close to the dimension of the original set such that the IFS satisfies strong separation condition (SSC) and the functions share a common contraction ratio. That is, the IFS is homogeneous. We show that under the above conditions there exists a homogeneous self-affine set satisfying the strong separation condition which approximates the dimension

• f the original set from below.
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Results Theorem D

Theorem D

For an IFS G = {Âi}M

i=1 we define the kth iterate by

Gk = {Âi1 ¶ · · · ¶ Âik}M

i1,...,ik=1 .

Theorem D Let Φ be an IFS of the form (1) and let Λ be the attractor of Φ. Without loss of generality we may assume that s— Æ s–. Suppose that either

1

Φ– satisfies the Hochman-condition and s– Æ 1,

• r

2

Φ–, Φ— satisfy the Hochman-condition and s— Æ 1 < s–.

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Results Theorem D

Theorem D (cont.)

Then for every Á > 0 there exists a homogeneous affine IFS Ψ of the form (8) Ψ = {Tj(x, y) = (–x + uj,1, —y + uj,2)}k

j=1

with attractor Γ ™ Λ such that Ψ is a subsystem of some iterate of Φ and satisfies the SSC, i.e. Ti(Γ) fl Tj(Γ) = ÿ and dimH Λ ≠ Á = dimP Λ ≠ Á = dimB Λ ≠ Á Æ dimH Γ = dimP Γ = dimB Γ .

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Feng-Hu Theorem about Ledrappier-Young formula

1

Introduction Abstract Notation An old result of B´ ar´ any Bal´ azs Hochman-condition

2

Results Theorem A Theorem B Proposition C Theorem D

3

Feng-Hu Theorem about Ledrappier-Young formula Notation The self-similar case Diagonally self-affine case

4

Stuff what I am going to have no time for

• B. B´

ar´ any, M. Rams, K. Simon Diagonally Self-affine sets and measures February 19, 2016 31 / 43

Feng-Hu Theorem about Ledrappier-Young formula Notation

Notation

First we recall here some results and notations of Feng and Hu. Let Ψ = {Âi}M

i=1

be a strictly contracting IFS mapping [0, 1]d into itself. Let Σ = {1, . . . , M}N be the corresponding symbolic space, ‡ the usual left-shift operator on Σ and let m be a ‡-invariant ergodic measure on Σ.

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Feng-Hu Theorem about Ledrappier-Young formula Notation

Notation (cont.)

Let Π be the natural projection, i.e. Π(i0, i1, . . . ) = limnæŒ Âi0 ¶ · · · ¶ Âin(0). Let P = {[1], . . . , [M]} be the partition of Σ, where [i] = {i œ Σ : i0 = i} and denote by B the Borel ‡-algebra

• f Rd.
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Feng-Hu Theorem about Ledrappier-Young formula The self-similar case

The projection entropy

We define the projection entropy

• f m under Π with

respect to Ψ as hΠ(m) := Hm(P | ‡≠1Π≠1B) ≠ Hm(P | Π≠1B) , where Hm(› | ÷) denotes the usual conditional entropy of › given ÷.

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Feng-Hu Theorem about Ledrappier-Young formula The self-similar case

Feng-Hu Theorem for self-similar IFS

Let Ψ be an IFS of similarities on the real line. Then dimH µ = hΠ(P) ‰ , where µ = P ¶ Π≠1 and ‰ = ≠

M ÿ i=1

pi log |ÂÕ

i(0)|

is the Lyapunov exponent.

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Feng-Hu Theorem about Ledrappier-Young formula Diagonally self-affine case

Notation

Let us assume that the maps of the IFS Ψ =

Ó

Âi : [0, 1]d ‘æ [0, 1]dÔM

i=1 have the form

Âi(x1, . . . , xd) = (fl1,ix1 + t1,i, . . . , fld,ixd + td,i). For a P = {p1, . . . , pM}N Bernoulli measure, denote the Lyapunov exponents by ‰j = ≠

M ÿ i=1

pi log |flj,i|. Without loss of generality we may assume that 0 < ‰1 Æ ‰2 Æ · · · Æ ‰d.

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Feng-Hu Theorem about Ledrappier-Young formula Diagonally self-affine case

Notation (cont.)

Let Ψk be the IFS with functions restricted to the first k coordinates, i.e. Ψk =

Ó

Âk

i : [0, 1]k ‘æ [0, 1]kÔM i=1 , where

Âk

i (x1, . . . , xk) = {(fl1,ix1 + t1,i, . . . , flk,ixk + tk,i)}M i=1 .

Denote the natural projection w.r.t Ψk by Πk. Moreover, let µk = P ¶ Π≠1

k ,

where P = (p1, . . . pk)N.

• B. B´

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Feng-Hu Theorem about Ledrappier-Young formula Diagonally self-affine case

Feng-Hu Theorem

For every 1 Æ k Æ d, dimH µk = hΠ1(P) ‰1 +

k ÿ j=2

hΠj(P) ≠ hΠj−1(P) ‰j . In particular, on the plane assuming SSC (9) dimH(µ) =

hµ ‰2(µ) + 3

1 ≠ ‰1(µ)

‰2(µ) 4

· dimH(µx) , where µx = projxµ. That is if SSC holds then (10) D(µ) = dimH(µ) ≈ ∆ dimH(µx) = min

;

1,

hµ ‰1(µ) <

.

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Stuff what I am going to have no time for

1

Introduction Abstract Notation An old result of B´ ar´ any Bal´ azs Hochman-condition

2

Results Theorem A Theorem B Proposition C Theorem D

3

Feng-Hu Theorem about Ledrappier-Young formula Notation The self-similar case Diagonally self-affine case

4

Stuff what I am going to have no time for

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Stuff what I am going to have no time for

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ar´ any: Dimension of the generalized 4-corner set and its projections, Erg. Th. & Dyn. Sys. 32 (2012), 1190-1215. [2]

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Mathematics, Vol. 34.

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