Magnification dynamics and non-conformality · 2013. 8. 10. · Spiritofthetalk “You take an obvious concept of a limit, and then, by the power of analysis, you can go to the limit

Magnification dynamics and non-conformality

Tuomas Sahlsten

XXII Rolf Nevanlinna Colloquium, August 2013Helsinki, Finland

joint work with Andrew Ferguson and Jonathan Fraser

x 2

Spirit of the talk

“You take an obvious concept of a limit, and then, by the power ofanalysis, you can go to the limit many times, which creates structures thatyou have not seen before. You think you have not done anything but,amazingly, you have achieved something.”

–Mikhail Gromov, March 2010

Magnification dynamics I: Developments

• Groundwork was set by Hillel Furstenberg already in the 70’s.It was only until 2008 when he popularised/axiomatised the notion.• Later developed by Gavish (2011), Hochman-Shmerkin (2012) and

Hochman (2010/2013).• Hochman-Shmerkin (2012): Applications to projection theorems,

which yielded a solution to a conjecture of Furstenberg (not thefamous ×2,×3 one however!)• Hochman-Shmerkin (2013): Applications to equidistribution problems

in metric number theory• ...more in progress ,


• Groundwork was set by Hillel Furstenberg already in the 70’s.It was only until 2008 when he popularised/axiomatised the notion.

• Later developed by Gavish (2011), Hochman-Shmerkin (2012) andHochman (2010/2013).• Hochman-Shmerkin (2012): Applications to projection theorems,





Hochman (2010/2013).

• Hochman-Shmerkin (2012): Applications to projection theorems,which yielded a solution to a conjecture of Furstenberg (not thefamous ×2,×3 one however!)• Hochman-Shmerkin (2013): Applications to equidistribution problems





which yielded a solution to a conjecture of Furstenberg (not thefamous ×2,×3 one however!)

• Hochman-Shmerkin (2013): Applications to equidistribution problemsin metric number theory• ...more in progress ,

Magnification dynamics II: ’Dyadic’ definition

• Let {Dk}k∈N be the filtration of Rd with dyadic cubes.• Let TD : Rd → Rd be the orientation preserving similitude that maps a

dyadic cube D onto [0, 1)d.• The magnification µD of a measure µ ∈ P([0, 1)d) to D withµ(D) > 0 is

µD =1

µ(D)TD∗(µ|D) ∈ P([0, 1)d)

• If k ∈ N, let Dk(x) ∈ Dk be the cube with x ∈ Dk(x). Write

Ξ = {(x, µ) : µ ∈ P([0, 1)d) and µ(Dk(x)) > 0 for any k ∈ N}.

• Define the magnification operator M : Ξ→ Ξ by

M(x, µ) = (TD1(x)(x), µD1(x))

on the phase space Ξ.


• Let {Dk}k∈N be the filtration of Rd with dyadic cubes.

• Let TD : Rd → Rd be the orientation preserving similitude that maps adyadic cube D onto [0, 1)d.• The magnification µD of a measure µ ∈ P([0, 1)d) to D withµ(D) > 0 is

µD =1

µ(D)TD∗(µ|D) ∈ P([0, 1)d)




M(x, µ) = (TD1(x)(x), µD1(x))




dyadic cube D onto [0, 1)d.

• The magnification µD of a measure µ ∈ P([0, 1)d) to D withµ(D) > 0 is

µD =1

µ(D)TD∗(µ|D) ∈ P([0, 1)d)




M(x, µ) = (TD1(x)(x), µD1(x))




dyadic cube D onto [0, 1)d.• The magnification µD of a measure µ ∈ P([0, 1)d) to D withµ(D) > 0 is

µD =1

µ(D)TD∗(µ|D) ∈ P([0, 1)d)




M(x, µ) = (TD1(x)(x), µD1(x))


Magnification dynamics III: invariant measures?

Iteration:Mk(x, µ) = (TDk(x)(x), µ

Dk(x)), (x, µ) ∈ Ξ.

• Let (x, µ) ∈ Ξ and N ∈ N. The scenery distribution of µ at x is

1

N

N−1∑k=0

δMk(x,µ) ∈ P(Ξ).

• A micromeasure distribution of µ at x is an accumulation point ofscenery distributions in P(Ξ) for weak topology.• The measure component a micromeasure distribution is supported onmicromeasures of µ at x (i.e. accumulation points of the’minimeasures’ µDk(x), k ∈ N)• Any micromeasure distribution Q is an M invariant measure i.e.M∗Q = Q.



Dk(x)), (x, µ) ∈ Ξ.


1

N

N−1∑k=0


• A micromeasure distribution of µ at x is an accumulation point ofscenery distributions in P(Ξ) for weak topology.

• The measure component a micromeasure distribution is supported onmicromeasures of µ at x (i.e. accumulation points of the’minimeasures’ µDk(x), k ∈ N)• Any micromeasure distribution Q is an M invariant measure i.e.M∗Q = Q.



Dk(x)), (x, µ) ∈ Ξ.


1

N

N−1∑k=0


• A micromeasure distribution of µ at x is an accumulation point ofscenery distributions in P(Ξ) for weak topology.• The measure component a micromeasure distribution is supported onmicromeasures of µ at x (i.e. accumulation points of the’minimeasures’ µDk(x), k ∈ N)

• Any micromeasure distribution Q is an M invariant measure i.e.M∗Q = Q.



Dk(x)), (x, µ) ∈ Ξ.


1

N

N−1∑k=0


• A micromeasure distribution of µ at x is an accumulation point ofscenery distributions in P(Ξ) for weak topology.• The measure component a micromeasure distribution is supported onmicromeasures of µ at x (i.e. accumulation points of the’minimeasures’ µDk(x), k ∈ N)• Any micromeasure distribution Q is an M invariant measure i.e.M∗Q = Q.

Magnification dynamics IV: ergodic CP distributions

Let Q̃ be the measure component of an M invariant measure Q.

• Q is a CP distribution, if the choice of (x, µ) according to Q can bemade by first choosing µ according to Q̃ and then x according to µ.I.e. we have a disintegration

ˆf(x, µ) dQ(x, µ) =

ˆˆf(x, µ) dµ(x) dQ̃(µ),

for any continuous f : Ξ→ R.• A CP distribution Q is (M -)ergodic if the Q measure of any M

invariant set is either 0 or 1. I.e. A = M−1A =⇒ Q(A) ∈ {0, 1}.• It is also possible to use more general (regular) filtrations than dyadic.

Then the dynamics is described by a Markov process (a CP chain)and the CP distribution is the stationary measure for this chain.



• Q is a CP distribution, if the choice of (x, µ) according to Q can bemade by first choosing µ according to Q̃ and then x according to µ.

I.e. we have a disintegrationˆf(x, µ) dQ(x, µ) =










for any continuous f : Ξ→ R.

• A CP distribution Q is (M -)ergodic if the Q measure of any Minvariant set is either 0 or 1. I.e. A = M−1A =⇒ Q(A) ∈ {0, 1}.• It is also possible to use more general (regular) filtrations than dyadic.








invariant set is either 0 or 1. I.e. A = M−1A =⇒ Q(A) ∈ {0, 1}.

• It is also possible to use more general (regular) filtrations than dyadic.Then the dynamics is described by a Markov process (a CP chain)and the CP distribution is the stationary measure for this chain.

Magnification dynamics V: generated CP distributions

A CP distribution Q is generated by a measure µ, if(1) Q is the only micromeasure distribution of µ at µ almost every x;(2) and at µ almost every x also the q-sparse scenery distributions

1

N

N−1∑k=0

δMqk(x,µ) ∈ P(Ξ)

converge to some, possibly other, distribution Qq for any q ∈ N.The condition (2) is technical, but is essential to carry geometricinformation from the micromeasure level back to µ.Often Qq = Q for any q ∈ N.


A CP distribution Q is generated by a measure µ, if(1) Q is the only micromeasure distribution of µ at µ almost every x;

(2) and at µ almost every x also the q-sparse scenery distributions

1

N

N−1∑k=0




A CP distribution Q is generated by a measure µ, if(1) Q is the only micromeasure distribution of µ at µ almost every x;(2) and at µ almost every x also the q-sparse scenery distributions

1

N

N−1∑k=0



Example: Distortion of dimension under projections• Let Πd,k be the set of all orthogonal projections Rd → Rk, k < d.

• For a CP distribution Q, write

E(π) =

ˆdimπ∗ν dQ̃(ν), π ∈ Πd,k.

i.e. the expected Hausdorff dimension of the projection π∗ν overmicromeasures ν.• The map E : Πd,k → R is lower semicontinuous.

Theorem (Hochman-Shmerkin 2012)

Suppose µ generates an ergodic CP distribution Q. Then(1) dimπ∗µ ≥ E(π) for any π ∈ Πd,k;(2) for any C1 map f : Rd → Rk without singular points

dim fµ ≥ ess infx∼µ

E(Dxf).

Moreover, for a.e. π ∈ Πd,k the expectation E(π) = min{k,dimµ}.

Example: Distortion of dimension under projections• Let Πd,k be the set of all orthogonal projections Rd → Rk, k < d.• For a CP distribution Q, write

E(π) =


i.e. the expected Hausdorff dimension of the projection π∗ν overmicromeasures ν.

• The map E : Πd,k → R is lower semicontinuous.




E(Dxf).



E(π) =





Suppose µ generates an ergodic CP distribution Q. Then(1) dimπ∗µ ≥ E(π) for any π ∈ Πd,k;

(2) for any C1 map f : Rd → Rk without singular points


E(Dxf).



E(π) =







E(Dxf).


Non-conformality I

• A C1 map f : R2 → R2 is ’non-conformal’ if the derivative matrixDxf has distinct singular values at any x ∈ R2.• Example: Fix integers m < n and define Tm,n : [0, 1]2 → [0, 1]2,

Tm,n(x, y) = (Tm(x), Tn(x)),

where Tm : [0, 1]→ [0, 1] is given by

Tm(x) = mx mod 1.

Here [0, 1] is thought as a unit circle T and [0, 1]2 as the 2-torus T2.

Non-conformality I

• A C1 map f : R2 → R2 is ’non-conformal’ if the derivative matrixDxf has distinct singular values at any x ∈ R2.

• Example: Fix integers m < n and define Tm,n : [0, 1]2 → [0, 1]2,

Tm,n(x, y) = (Tm(x), Tn(x)),


Tm(x) = mx mod 1.


Non-conformality I

• A C1 map f : R2 → R2 is ’non-conformal’ if the derivative matrixDxf has distinct singular values at any x ∈ R2.• Example: Fix integers m < n and define Tm,n : [0, 1]2 → [0, 1]2,

Tm,n(x, y) = (Tm(x), Tn(x)),


Tm(x) = mx mod 1.


Non-conformality II: Tm,n invariant sets

A subset K ⊂ [0, 1]2 is a Bedford-McMullen carpet, if K =⋃i,j fi,j(K),

where

fi,j =

(1/m 0

0 1/n

)+

(ij

),

for some indices 0 ≤ i ≤ m− 1 and 0 ≤ j ≤ n− 1. Then K = T−1m,n(K).

Non-conformality II: Tm,n invariant measures

• A Tm,n invariant measure µ on [0, 1]2 is Bernoulli if all ’constructionrectangles’ are mutually µ independent. I.e.

µ[(I × J) ∩ (I ′ × J ′)] = µ(I × J)µ(I ′ × J ′)

for all m-adic intervals I, I ′ ⊂ [0, 1] and n-adic intervals J, J ′ ⊂ [0, 1].• Example: The McMullen measure of a Bedford-McMullen carpet is

a Tm,n invariant Bernoulli measure of maximal Hausdorff dimension.



µ[(I × J) ∩ (I ′ × J ′)] = µ(I × J)µ(I ′ × J ′)

for all m-adic intervals I, I ′ ⊂ [0, 1] and n-adic intervals J, J ′ ⊂ [0, 1].

• Example: The McMullen measure of a Bedford-McMullen carpet isa Tm,n invariant Bernoulli measure of maximal Hausdorff dimension.



µ[(I × J) ∩ (I ′ × J ′)] = µ(I × J)µ(I ′ × J ′)

for all m-adic intervals I, I ′ ⊂ [0, 1] and n-adic intervals J, J ′ ⊂ [0, 1].• Example: The McMullen measure of a Bedford-McMullen carpet is

a Tm,n invariant Bernoulli measure of maximal Hausdorff dimension.

Non-conformality III: Magnification

• It was proved by Käenmäki and Bandt (2011) that under mildassumptions the ’tangent sets’ of Bedford-McMullen carpets (wrt.Hausdorff distance) are of the form

[0, 1]× C,

where C is some random Cantor set.• The product form of the tangents was exploited by Mackay (2011)

when computing the conformal Assouad dimension ofBedford-McMullen carpets.



[0, 1]× C,

where C is some random Cantor set.

• The product form of the tangents was exploited by Mackay (2011)when computing the conformal Assouad dimension ofBedford-McMullen carpets.



[0, 1]× C,

where C is some random Cantor set.• The product form of the tangents was exploited by Mackay (2011)

when computing the conformal Assouad dimension ofBedford-McMullen carpets.

Non-conformality III: Magnification, cont.

TheoremAny Tm,n Bernoulli measure µ generates an ergodic CP distribution Q.

• Measure component Q̃ is the distribution of the random measure

π1µ× µx,

where x ∼ π1µ and µx ∈ P([0, 1]) is the conditional measure of µwith respect to the fibre π−11 {x}.• Here we do not use dyadic cubes to magnify but more general but stillregular partitions obtained from ’construction rectangles’ I ×J , whereI, J ⊂ [0, 1] are m-adic and n-adic intervals respectively. Hence theCP distribution Q is a stationary measure for an ergodic CP chain.

Application I: Projections

Furstenberg’s Conjecture (one of them, from 1960’s)

If X,Y ⊂ [0, 1] are closed and T2 and T3 invariant respectively. Then

dimπ(X × Y ) = min{1,dim(X × Y )}, π ∈ Π2,1 \ {π1, π2}.

Solved:


If µ, ν ∈ P([0, 1]) are Tm and Tn invariant respectively and logmlogn ∈ R \Q,then

dimπ∗(µ× ν) = min{1,dim(µ× ν)}, π ∈ Π2,1 \ {π1, π2}.

Obtained by constructing an ergodic CP distribution out of µ× ν.

Application I: Projections

Furstenberg’s Conjecture (one of them, from 1960’s)

If X,Y ⊂ [0, 1] are closed and T2 and T3 invariant respectively. Then

dimπ(X × Y ) = min{1,dim(X × Y )}, π ∈ Π2,1 \ {π1, π2}.

Solved:


If µ, ν ∈ P([0, 1]) are Tm and Tn invariant respectively and logmlogn ∈ R \Q,then

dimπ∗(µ× ν) = min{1, dim(µ× ν)}, π ∈ Π2,1 \ {π1, π2}.

Obtained by constructing an ergodic CP distribution out of µ× ν.

Application I: Projections, cont.

Conjecture

Suppose µ is a Tm,n invariant measure and logmlogn ∈ R \Q, then

dimπ∗µ = min{1, dimµ}, π ∈ Π2,1 \ {π1, π2}.

Theorem (Ferguson-Jordan-Shmerkin 2010)

Suppose K is a Bedford-McMullen carpet with logmlogn ∈ R \Q. Then

dimπ(K) = min{1,dimK}, π ∈ Π2,1 \ {π1, π2}.

TheoremThe conjecture above holds for Tm,n invariant Bernoulli measures.

Application I: Projections, cont.

Conjecture

Suppose µ is a Tm,n invariant measure and logmlogn ∈ R \Q, then

dimπ∗µ = min{1, dimµ}, π ∈ Π2,1 \ {π1, π2}.

Theorem (Ferguson-Jordan-Shmerkin 2010)

Suppose K is a Bedford-McMullen carpet with logmlogn ∈ R \Q. Then

dimπ(K) = min{1, dimK}, π ∈ Π2,1 \ {π1, π2}.

TheoremThe conjecture above holds for Tm,n invariant Bernoulli measures.




E(Dxf).

Moreover, for a.e. π ∈ Πd,k the value E(π) = min{k, dimµ}.

• After suitable reparametrisation of Π2,1, the map E is invariant underthe irrational logmlogn rotation of the circle, so E is constant as a lowersemicontinuous function.

Application II: Distance sets

The distance set of K ⊂ Rd is

D(K) = {|x− y| : x, y ∈ K}.

Distance set conjecture (Falconer, 1980’s)

Suppose K ⊂ Rd is Borel and dimK ≥ d/2. Then dimD(K) = 1.Moreover, if dimK > d/2, then L1(D(K)) > 0.

Many people involved, current state:• Bourgain (2003) found a small constant ε > 0 with

dimD(K) ≥ 12

+ ε

whenever K ⊂ R2 with dimK ≥ 1.• Erdogan (2006) proved dimK > d/2 + 1/3 in Rd yields positive

measure for D(K).• Orponen (2011) proved dimD(K) = 1 if K is a planar self-similar set

with H1(K) > 0.

Application II: Distance setsThe distance set of K ⊂ Rd is

D(K) = {|x− y| : x, y ∈ K}.

Distance set conjecture (Falconer, 1980’s)

Suppose K ⊂ Rd is Borel and dimK ≥ d/2. Then dimD(K) = 1.Moreover, if dimK > d/2, then L1(D(K)) > 0.

Many people involved, current state:• Bourgain (2003) found a small constant ε > 0 with

dimD(K) ≥ 12

+ ε

whenever K ⊂ R2 with dimK ≥ 1.• Erdogan (2006) proved dimK > d/2 + 1/3 in Rd yields positive

measure for D(K).• Orponen (2011) proved dimD(K) = 1 if K is a planar self-similar set

with H1(K) > 0.

Application II: Distance sets, cont.

TheoremIf µ on R2 generates an ergodic CP distribution and H1(sptµ) > 0, then

dimD(sptµ) ≥ min{1,dimµ}.

Corollary

If µ is any measure on R2 with dimµ > 1, then after a random translationof µ, there are micromeasures ν of µ at µ a.e. x with dimD(spt ν) = 1.

Corollary

If K is a Bedford-McMullen carpet with dimK ≥ 1, then dimD(K) = 1.

• Using standard dimension approximation theorems viaBedford-McMullen ’type’ carpets, this yields results for otherLalley-Gatzouras and Barański type self-affine carpets as well.

Possible further topics

• Scaling scenery of more general Tm,n invariant measures(Gibbs measures with strong mixing properties to replace Bernoulli)• Limit geometry of general self-affine/other non-conformal sets• Projection theorems in the Heisenberg group• CP distributions in metric spaces (Gromov-Hausdorff, Christ?)• Conformal Hausdorff dimension of self-affine carpets (???)


• Scaling scenery of more general Tm,n invariant measures(Gibbs measures with strong mixing properties to replace Bernoulli)

• Limit geometry of general self-affine/other non-conformal sets• Projection theorems in the Heisenberg group• CP distributions in metric spaces (Gromov-Hausdorff, Christ?)• Conformal Hausdorff dimension of self-affine carpets (???)


• Scaling scenery of more general Tm,n invariant measures(Gibbs measures with strong mixing properties to replace Bernoulli)• Limit geometry of general self-affine/other non-conformal sets

• Projection theorems in the Heisenberg group• CP distributions in metric spaces (Gromov-Hausdorff, Christ?)• Conformal Hausdorff dimension of self-affine carpets (???)


• Scaling scenery of more general Tm,n invariant measures(Gibbs measures with strong mixing properties to replace Bernoulli)• Limit geometry of general self-affine/other non-conformal sets• Projection theorems in the Heisenberg group

• CP distributions in metric spaces (Gromov-Hausdorff, Christ?)• Conformal Hausdorff dimension of self-affine carpets (???)


• Scaling scenery of more general Tm,n invariant measures(Gibbs measures with strong mixing properties to replace Bernoulli)• Limit geometry of general self-affine/other non-conformal sets• Projection theorems in the Heisenberg group• CP distributions in metric spaces (Gromov-Hausdorff, Christ?)

• Conformal Hausdorff dimension of self-affine carpets (???)


• Scaling scenery of more general Tm,n invariant measures(Gibbs measures with strong mixing properties to replace Bernoulli)• Limit geometry of general self-affine/other non-conformal sets• Projection theorems in the Heisenberg group• CP distributions in metric spaces (Gromov-Hausdorff, Christ?)• Conformal Hausdorff dimension of self-affine carpets (???)

ReferencesC. Bandt, A. Käenmäki: Local structure of self-affine sets (2011), ErgodicTheory and Dynamical Systems, to appear

A. Ferguson, J. Fraser, T. S.: Scaling scenery of non-conformal measures(2013), preprint, 25 pages, arXiv:1307.5023

A. Ferguson, T. Jordan, P. Shmerkin: The Hausdorff dimension of theprojections of self-affine carpets (2010), Fundamenta Mathematicae 209, 193–213

H. Furstenberg: Ergodic fractal measures and dimension conservation (2008),Ergodic Theory and Dynamical Systems 28, 405–422

M. Gavish: Measures with uniform scaling scenery (2011), Ergodic Theory andDynamical Systems 31, 33–48

M. Hochman: Dynamics on fractals and fractal distributions (2010/2013),preprint, 62 pages, arXiv:1008.3731

M. Hochman, P. Shmerkin: Local entropy averages and projections of fractalmeasures (2012), Annals of Mathematics (2), 175(3):1001–1059

M. Hochman, P. Shmerkin: Equidistribution from fractals (2013), preprint,44 pages, arXiv:1302.5792

J. Mackay: Assouad dimension of self-affine carpets (2011), ConformalGeometry and Dynamics 15, 177–187

Kiitos!

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Magnification dynamics and non-conformality · 2013. 8. 10. · Spiritofthetalk “You take an obvious concept of a limit, and then, by the power of analysis, you can go to the limit