# Local entropy averages and projections of fractal measures

```					Background            Motivation and results         Outline of proof          Ergodic self-similarity

Local entropy averages and projections of
fractal measures

Pablo Shmerkin (joint work with M. Hochman)

Centre for Interdisciplinary Computational And Dynamical Analysis
and School of Mathematics
University of Manchester, UK

St Andrews, 5 November 2009
Background           Motivation and results   Outline of proof    Ergodic self-similarity

Sets and measures

In what follows, there are parallel developments for sets and
measures. To avoid stating everything twice, we will focus on
measures, as:
Results for measures imply the corresponding results for
sets in all the cases we consider.
Our methods work naturally in the measure setting.
Background           Motivation and results   Outline of proof    Ergodic self-similarity

Sets and measures

In what follows, there are parallel developments for sets and
measures. To avoid stating everything twice, we will focus on
measures, as:
Results for measures imply the corresponding results for
sets in all the cases we consider.
Our methods work naturally in the measure setting.
Background           Motivation and results   Outline of proof    Ergodic self-similarity

Sets and measures

In what follows, there are parallel developments for sets and
measures. To avoid stating everything twice, we will focus on
measures, as:
Results for measures imply the corresponding results for
sets in all the cases we consider.
Our methods work naturally in the measure setting.
Background            Motivation and results         Outline of proof           Ergodic self-similarity

Local dimension

Question
How do we “know” that d-dimensional Lebesgue measure is
d-dimensional? In general, how do we deﬁne the dimension of
a general (possibly fractal) measure?

Do it locally: if µ(B(x, r )) scales like r α as r                0, then we say
that the local dimension of µ at x is α.

log µ(B(x,r ))
Formally, dim(µ, x) = limr         0      log r     (If deﬁned).
For Lebesgue measure λ            on Rd , dim(λ, x) = d for all        x.
Background            Motivation and results         Outline of proof           Ergodic self-similarity

Local dimension

Question
How do we “know” that d-dimensional Lebesgue measure is
d-dimensional? In general, how do we deﬁne the dimension of
a general (possibly fractal) measure?

Do it locally: if µ(B(x, r )) scales like r α as r                0, then we say
that the local dimension of µ at x is α.

log µ(B(x,r ))
Formally, dim(µ, x) = limr         0      log r     (If deﬁned).
For Lebesgue measure λ            on Rd , dim(λ, x) = d for all        x.
Background            Motivation and results         Outline of proof           Ergodic self-similarity

Local dimension

Question
How do we “know” that d-dimensional Lebesgue measure is
d-dimensional? In general, how do we deﬁne the dimension of
a general (possibly fractal) measure?

Do it locally: if µ(B(x, r )) scales like r α as r                0, then we say
that the local dimension of µ at x is α.

log µ(B(x,r ))
Formally, dim(µ, x) = limr         0      log r     (If deﬁned).
For Lebesgue measure λ            on Rd , dim(λ, x) = d for all        x.
Background            Motivation and results         Outline of proof           Ergodic self-similarity

Local dimension

Question
How do we “know” that d-dimensional Lebesgue measure is
d-dimensional? In general, how do we deﬁne the dimension of
a general (possibly fractal) measure?

Do it locally: if µ(B(x, r )) scales like r α as r                0, then we say
that the local dimension of µ at x is α.

log µ(B(x,r ))
Formally, dim(µ, x) = limr         0      log r     (If deﬁned).
For Lebesgue measure λ            on Rd , dim(λ, x) = d for all        x.
Background         Motivation and results   Outline of proof   Ergodic self-similarity

Exact dimension

In general, it is too much to ask that the local dimension exists
and has the same value at all points of the support. But for
many natural measures, the local dimension exists and has the
same value at almost all points (relative to the measure we are
analyzing).
Deﬁnition
A measure µ is exact dimensional if dim(µ, x) exists and is
almost everywhere constant. This constant value α is the exact
dimension of µ, denoted dim µ = α.
Background         Motivation and results   Outline of proof   Ergodic self-similarity

Exact dimension

In general, it is too much to ask that the local dimension exists
and has the same value at all points of the support. But for
many natural measures, the local dimension exists and has the
same value at almost all points (relative to the measure we are
analyzing).
Deﬁnition
A measure µ is exact dimensional if dim(µ, x) exists and is
almost everywhere constant. This constant value α is the exact
dimension of µ, denoted dim µ = α.
Background            Motivation and results           Outline of proof    Ergodic self-similarity

Various notions of dimension

In general, local dimension is not deﬁned, and even if it is
its value can vary a lot.
We deﬁne the lower local dimension as
log µ(B(x, r ))
dim(µ, x) = lim inf                       ,
r   0        log r

and likewise we deﬁne upper local dimension.
Then we can globalize by taking the essential inﬁmum, the
essential supremum, or the average of the values of the
lower/upper local dimension.
Many notions of dimension! In general they can all be
different. They all have names and alternative deﬁnitions,
but we will need just one of them.
Background            Motivation and results           Outline of proof    Ergodic self-similarity

Various notions of dimension

In general, local dimension is not deﬁned, and even if it is
its value can vary a lot.
We deﬁne the lower local dimension as
log µ(B(x, r ))
dim(µ, x) = lim inf                       ,
r   0        log r

and likewise we deﬁne upper local dimension.
Then we can globalize by taking the essential inﬁmum, the
essential supremum, or the average of the values of the
lower/upper local dimension.
Many notions of dimension! In general they can all be
different. They all have names and alternative deﬁnitions,
but we will need just one of them.
Background            Motivation and results           Outline of proof    Ergodic self-similarity

Various notions of dimension

In general, local dimension is not deﬁned, and even if it is
its value can vary a lot.
We deﬁne the lower local dimension as
log µ(B(x, r ))
dim(µ, x) = lim inf                       ,
r   0        log r

and likewise we deﬁne upper local dimension.
Then we can globalize by taking the essential inﬁmum, the
essential supremum, or the average of the values of the
lower/upper local dimension.
Many notions of dimension! In general they can all be
different. They all have names and alternative deﬁnitions,
but we will need just one of them.
Background            Motivation and results           Outline of proof    Ergodic self-similarity

Various notions of dimension

In general, local dimension is not deﬁned, and even if it is
its value can vary a lot.
We deﬁne the lower local dimension as
log µ(B(x, r ))
dim(µ, x) = lim inf                       ,
r   0        log r

and likewise we deﬁne upper local dimension.
Then we can globalize by taking the essential inﬁmum, the
essential supremum, or the average of the values of the
lower/upper local dimension.
Many notions of dimension! In general they can all be
different. They all have names and alternative deﬁnitions,
but we will need just one of them.
Background           Motivation and results     Outline of proof   Ergodic self-similarity

lower Hausdorff dimension

Out of all the alternatives, we will be concerned with the
smallest of all.
Deﬁnition
The lower Hausdorff dimension of a measure µ is deﬁned as

dim∗ µ = essinf dim(µ, x).

Remarks
If (and only if) µ is exact dimensional, dim µ = dim∗ µ.
Morally speaking, dim∗ (µ) is the smallest number α such
that µ(B(x, r )) ≤ r α for typical x and small r .
Background           Motivation and results     Outline of proof   Ergodic self-similarity

lower Hausdorff dimension

Out of all the alternatives, we will be concerned with the
smallest of all.
Deﬁnition
The lower Hausdorff dimension of a measure µ is deﬁned as

dim∗ µ = essinf dim(µ, x).

Remarks
If (and only if) µ is exact dimensional, dim µ = dim∗ µ.
Morally speaking, dim∗ (µ) is the smallest number α such
that µ(B(x, r )) ≤ r α for typical x and small r .
Background           Motivation and results     Outline of proof   Ergodic self-similarity

lower Hausdorff dimension

Out of all the alternatives, we will be concerned with the
smallest of all.
Deﬁnition
The lower Hausdorff dimension of a measure µ is deﬁned as

dim∗ µ = essinf dim(µ, x).

Remarks
If (and only if) µ is exact dimensional, dim µ = dim∗ µ.
Morally speaking, dim∗ (µ) is the smallest number α such
that µ(B(x, r )) ≤ r α for typical x and small r .
Background           Motivation and results     Outline of proof   Ergodic self-similarity

lower Hausdorff dimension

Out of all the alternatives, we will be concerned with the
smallest of all.
Deﬁnition
The lower Hausdorff dimension of a measure µ is deﬁned as

dim∗ µ = essinf dim(µ, x).

Remarks
If (and only if) µ is exact dimensional, dim µ = dim∗ µ.
Morally speaking, dim∗ (µ) is the smallest number α such
that µ(B(x, r )) ≤ r α for typical x and small r .
Background           Motivation and results     Outline of proof   Ergodic self-similarity

lower Hausdorff dimension

Out of all the alternatives, we will be concerned with the
smallest of all.
Deﬁnition
The lower Hausdorff dimension of a measure µ is deﬁned as

dim∗ µ = essinf dim(µ, x).

Remarks
If (and only if) µ is exact dimensional, dim µ = dim∗ µ.
Morally speaking, dim∗ (µ) is the smallest number α such
that µ(B(x, r )) ≤ r α for typical x and small r .
Background            Motivation and results           Outline of proof   Ergodic self-similarity

Projecting measures

If µ is a measure on a space X and f : X → Y is a map, we
deﬁne the image measure f µ as

(f µ)(A) = µ(f −1 A).

In the case in which X = R2 , Y is a line and f : X → Y is
the orthogonal projection onto Y , (f µ)(B(y , r )) is the
measure of a stripe of width 2r orthogonal to Y .
Thus, in this case (f µ)(B(y , r )) is a global quantity (or at
least not local).
Background            Motivation and results           Outline of proof   Ergodic self-similarity

Projecting measures

If µ is a measure on a space X and f : X → Y is a map, we
deﬁne the image measure f µ as

(f µ)(A) = µ(f −1 A).

In the case in which X = R2 , Y is a line and f : X → Y is
the orthogonal projection onto Y , (f µ)(B(y , r )) is the
measure of a stripe of width 2r orthogonal to Y .
Thus, in this case (f µ)(B(y , r )) is a global quantity (or at
least not local).
Background            Motivation and results           Outline of proof   Ergodic self-similarity

Projecting measures

If µ is a measure on a space X and f : X → Y is a map, we
deﬁne the image measure f µ as

(f µ)(A) = µ(f −1 A).

In the case in which X = R2 , Y is a line and f : X → Y is
the orthogonal projection onto Y , (f µ)(B(y , r )) is the
measure of a stripe of width 2r orthogonal to Y .
Thus, in this case (f µ)(B(y , r )) is a global quantity (or at
least not local).
Background            Motivation and results   Outline of proof     Ergodic self-similarity

Projections and dimension I

Let π : R2 → be an orthogonal projection, and let µ be a
measure on R2 .
Even if µ is exact dimensional, there is a priori no reason
why πµ should be.
However, if y = πx, then B(x, r )) ⊂ π −1 (B(πx, r )), so
µ(B(x, r )) ≤ (πµ)(B(πx, r )).
It follows that dim∗ (πµ) ≤ dim∗ (µ) (larger measure=smaller
dimension).
Also, dim∗ (πµ) ≤ 1 (measures on a line can’t have
dimension more than 1).
Background            Motivation and results   Outline of proof     Ergodic self-similarity

Projections and dimension I

Let π : R2 → be an orthogonal projection, and let µ be a
measure on R2 .
Even if µ is exact dimensional, there is a priori no reason
why πµ should be.
However, if y = πx, then B(x, r )) ⊂ π −1 (B(πx, r )), so
µ(B(x, r )) ≤ (πµ)(B(πx, r )).
It follows that dim∗ (πµ) ≤ dim∗ (µ) (larger measure=smaller
dimension).
Also, dim∗ (πµ) ≤ 1 (measures on a line can’t have
dimension more than 1).
Background            Motivation and results   Outline of proof     Ergodic self-similarity

Projections and dimension I

Let π : R2 → be an orthogonal projection, and let µ be a
measure on R2 .
Even if µ is exact dimensional, there is a priori no reason
why πµ should be.
However, if y = πx, then B(x, r )) ⊂ π −1 (B(πx, r )), so
µ(B(x, r )) ≤ (πµ)(B(πx, r )).
It follows that dim∗ (πµ) ≤ dim∗ (µ) (larger measure=smaller
dimension).
Also, dim∗ (πµ) ≤ 1 (measures on a line can’t have
dimension more than 1).
Background            Motivation and results   Outline of proof     Ergodic self-similarity

Projections and dimension I

Let π : R2 → be an orthogonal projection, and let µ be a
measure on R2 .
Even if µ is exact dimensional, there is a priori no reason
why πµ should be.
However, if y = πx, then B(x, r )) ⊂ π −1 (B(πx, r )), so
µ(B(x, r )) ≤ (πµ)(B(πx, r )).
It follows that dim∗ (πµ) ≤ dim∗ (µ) (larger measure=smaller
dimension).
Also, dim∗ (πµ) ≤ 1 (measures on a line can’t have
dimension more than 1).
Background            Motivation and results   Outline of proof     Ergodic self-similarity

Projections and dimension I

Let π : R2 → be an orthogonal projection, and let µ be a
measure on R2 .
Even if µ is exact dimensional, there is a priori no reason
why πµ should be.
However, if y = πx, then B(x, r )) ⊂ π −1 (B(πx, r )), so
µ(B(x, r )) ≤ (πµ)(B(πx, r )).
It follows that dim∗ (πµ) ≤ dim∗ (µ) (larger measure=smaller
dimension).
Also, dim∗ (πµ) ≤ 1 (measures on a line can’t have
dimension more than 1).
Background           Motivation and results   Outline of proof   Ergodic self-similarity

Projections and dimension II

Remarks
We have seen that for a ﬁxed orthogonal projection, π,
dim∗ (πµ) ≤ min(dim∗ µ, 1).
In general, the inequality can be strict: let µ be Lebesgue
measure on a line and π orthogonal projection onto a line
orthogonal to . Then dim µ = 1 but πµ is a delta mass so
dim(πµ) = 0.
Orthogonal projections can be identiﬁed with their image,
which are lines through the origin, and lines through the
origin are parametrized by the angle they make with the
x-axis. So we can identify π with θ ∈ [0, π) =⇒ there are
a measure and a metric on the family of projections.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

Projections and dimension II

Remarks
We have seen that for a ﬁxed orthogonal projection, π,
dim∗ (πµ) ≤ min(dim∗ µ, 1).
In general, the inequality can be strict: let µ be Lebesgue
measure on a line and π orthogonal projection onto a line
orthogonal to . Then dim µ = 1 but πµ is a delta mass so
dim(πµ) = 0.
Orthogonal projections can be identiﬁed with their image,
which are lines through the origin, and lines through the
origin are parametrized by the angle they make with the
x-axis. So we can identify π with θ ∈ [0, π) =⇒ there are
a measure and a metric on the family of projections.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

Projections and dimension II

Remarks
We have seen that for a ﬁxed orthogonal projection, π,
dim∗ (πµ) ≤ min(dim∗ µ, 1).
In general, the inequality can be strict: let µ be Lebesgue
measure on a line and π orthogonal projection onto a line
orthogonal to . Then dim µ = 1 but πµ is a delta mass so
dim(πµ) = 0.
Orthogonal projections can be identiﬁed with their image,
which are lines through the origin, and lines through the
origin are parametrized by the angle they make with the
x-axis. So we can identify π with θ ∈ [0, π) =⇒ there are
a measure and a metric on the family of projections.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

Projections and dimension II

Remarks
We have seen that for a ﬁxed orthogonal projection, π,
dim∗ (πµ) ≤ min(dim∗ µ, 1).
In general, the inequality can be strict: let µ be Lebesgue
measure on a line and π orthogonal projection onto a line
orthogonal to . Then dim µ = 1 but πµ is a delta mass so
dim(πµ) = 0.
Orthogonal projections can be identiﬁed with their image,
which are lines through the origin, and lines through the
origin are parametrized by the angle they make with the
x-axis. So we can identify π with θ ∈ [0, π) =⇒ there are
a measure and a metric on the family of projections.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

Projections and dimension II

Remarks
We have seen that for a ﬁxed orthogonal projection, π,
dim∗ (πµ) ≤ min(dim∗ µ, 1).
In general, the inequality can be strict: let µ be Lebesgue
measure on a line and π orthogonal projection onto a line
orthogonal to . Then dim µ = 1 but πµ is a delta mass so
dim(πµ) = 0.
Orthogonal projections can be identiﬁed with their image,
which are lines through the origin, and lines through the
origin are parametrized by the angle they make with the
x-axis. So we can identify π with θ ∈ [0, π) =⇒ there are
a measure and a metric on the family of projections.
Background           Motivation and results    Outline of proof   Ergodic self-similarity

The projection theorem

Theorem (Marstrand, Kaufman, Mattila, Hunt-Kaloshin, . . .)
Let µ be a measure on R2 . Then

dim∗ (πµ) = min(dim∗ µ, 1)

for almost every orthogonal projection π.

Remarks
If dim∗ (πµ) < min(dim∗ µ, 1), we say that π is an
exceptional projection.
The theorem says nothing about the structure of the set of
exceptional projections. Its topology can be very
complicated.
Background           Motivation and results    Outline of proof   Ergodic self-similarity

The projection theorem

Theorem (Marstrand, Kaufman, Mattila, Hunt-Kaloshin, . . .)
Let µ be a measure on R2 . Then

dim∗ (πµ) = min(dim∗ µ, 1)

for almost every orthogonal projection π.

Remarks
If dim∗ (πµ) < min(dim∗ µ, 1), we say that π is an
exceptional projection.
The theorem says nothing about the structure of the set of
exceptional projections. Its topology can be very
complicated.
Background           Motivation and results    Outline of proof   Ergodic self-similarity

The projection theorem

Theorem (Marstrand, Kaufman, Mattila, Hunt-Kaloshin, . . .)
Let µ be a measure on R2 . Then

dim∗ (πµ) = min(dim∗ µ, 1)

for almost every orthogonal projection π.

Remarks
If dim∗ (πµ) < min(dim∗ µ, 1), we say that π is an
exceptional projection.
The theorem says nothing about the structure of the set of
exceptional projections. Its topology can be very
complicated.
Background           Motivation and results    Outline of proof   Ergodic self-similarity

The projection theorem

Theorem (Marstrand, Kaufman, Mattila, Hunt-Kaloshin, . . .)
Let µ be a measure on R2 . Then

dim∗ (πµ) = min(dim∗ µ, 1)

for almost every orthogonal projection π.

Remarks
If dim∗ (πµ) < min(dim∗ µ, 1), we say that π is an
exceptional projection.
The theorem says nothing about the structure of the set of
exceptional projections. Its topology can be very
complicated.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

A general question

General Question
If the measure µ has structure (e.g. arithmetical, dynamical),
can one determine the exact set of exceptions in the projection
theorem?

This problem was considered untractable even in the
simplest (nontrival) cases until recently.
Recent progress achieved by G. Moreira (different setting
but relevant techniques), Y. Peres-P. S. ,
F.Nazarov-Y.Peres-P.S., A. Ferguson, T. Jordan and P.S.,
and M. Hochman-P.S.
These results were motivated by questions of H.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

A general question

General Question
If the measure µ has structure (e.g. arithmetical, dynamical),
can one determine the exact set of exceptions in the projection
theorem?

This problem was considered untractable even in the
simplest (nontrival) cases until recently.
Recent progress achieved by G. Moreira (different setting
but relevant techniques), Y. Peres-P. S. ,
F.Nazarov-Y.Peres-P.S., A. Ferguson, T. Jordan and P.S.,
and M. Hochman-P.S.
These results were motivated by questions of H.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

A general question

General Question
If the measure µ has structure (e.g. arithmetical, dynamical),
can one determine the exact set of exceptions in the projection
theorem?

This problem was considered untractable even in the
simplest (nontrival) cases until recently.
Recent progress achieved by G. Moreira (different setting
but relevant techniques), Y. Peres-P. S. ,
F.Nazarov-Y.Peres-P.S., A. Ferguson, T. Jordan and P.S.,
and M. Hochman-P.S.
These results were motivated by questions of H.
Background           Motivation and results   Outline of proof   Ergodic self-similarity

A general question

General Question
If the measure µ has structure (e.g. arithmetical, dynamical),
can one determine the exact set of exceptions in the projection
theorem?

This problem was considered untractable even in the
simplest (nontrival) cases until recently.
Recent progress achieved by G. Moreira (different setting
but relevant techniques), Y. Peres-P. S. ,
F.Nazarov-Y.Peres-P.S., A. Ferguson, T. Jordan and P.S.,
and M. Hochman-P.S.
These results were motivated by questions of H.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Hillel Furstenberg

contributions to ergodic theory and its applications to other
areas.
Not so many papers (around 60 over a >50 year career)
but most of them were revolutionary.
Winner of Wolf Prize in 2006/7.
Perhaps most famous contribution is an ergodic-theoretic
proof of Szemerédi’s Theorem: if a set E ⊂ Z has positive
density, then it contains arbitrarily long arithmetic
progressions.
His (old) ideas on fractals were published in 2008. Many
deep insights that we exploit in our work.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Hillel Furstenberg

contributions to ergodic theory and its applications to other
areas.
Not so many papers (around 60 over a >50 year career)
but most of them were revolutionary.
Winner of Wolf Prize in 2006/7.
Perhaps most famous contribution is an ergodic-theoretic
proof of Szemerédi’s Theorem: if a set E ⊂ Z has positive
density, then it contains arbitrarily long arithmetic
progressions.
His (old) ideas on fractals were published in 2008. Many
deep insights that we exploit in our work.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Hillel Furstenberg

contributions to ergodic theory and its applications to other
areas.
Not so many papers (around 60 over a >50 year career)
but most of them were revolutionary.
Winner of Wolf Prize in 2006/7.
Perhaps most famous contribution is an ergodic-theoretic
proof of Szemerédi’s Theorem: if a set E ⊂ Z has positive
density, then it contains arbitrarily long arithmetic
progressions.
His (old) ideas on fractals were published in 2008. Many
deep insights that we exploit in our work.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Hillel Furstenberg

contributions to ergodic theory and its applications to other
areas.
Not so many papers (around 60 over a >50 year career)
but most of them were revolutionary.
Winner of Wolf Prize in 2006/7.
Perhaps most famous contribution is an ergodic-theoretic
proof of Szemerédi’s Theorem: if a set E ⊂ Z has positive
density, then it contains arbitrarily long arithmetic
progressions.
His (old) ideas on fractals were published in 2008. Many
deep insights that we exploit in our work.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Hillel Furstenberg

contributions to ergodic theory and its applications to other
areas.
Not so many papers (around 60 over a >50 year career)
but most of them were revolutionary.
Winner of Wolf Prize in 2006/7.
Perhaps most famous contribution is an ergodic-theoretic
proof of Szemerédi’s Theorem: if a set E ⊂ Z has positive
density, then it contains arbitrarily long arithmetic
progressions.
His (old) ideas on fractals were published in 2008. Many
deep insights that we exploit in our work.
Background   Motivation and results   Outline of proof   Ergodic self-similarity

Hillel Furstenberg
Background            Motivation and results      Outline of proof   Ergodic self-similarity

Invariant sets and measures

Deﬁnition
Let X be a space and T : X → X a transformation.
A set E ⊂ X is invariant if TE ⊂ E.
A measure µ on X is invariant if T µ = µ or in other words,
µ(T −1 A) = µ(A) for all measurable sets A.

Example (Morphisms of the circle)
Let S 1 = R/Z = [0, 1] with endpoints identiﬁed. For m ≥ 2 let
Tm : S 1 → S 1 be the “times m” map:

Tm (x) = m · x mod 1.

These are the simplest examples of chaotic transformations.
Background            Motivation and results      Outline of proof   Ergodic self-similarity

Invariant sets and measures

Deﬁnition
Let X be a space and T : X → X a transformation.
A set E ⊂ X is invariant if TE ⊂ E.
A measure µ on X is invariant if T µ = µ or in other words,
µ(T −1 A) = µ(A) for all measurable sets A.

Example (Morphisms of the circle)
Let S 1 = R/Z = [0, 1] with endpoints identiﬁed. For m ≥ 2 let
Tm : S 1 → S 1 be the “times m” map:

Tm (x) = m · x mod 1.

These are the simplest examples of chaotic transformations.
Background            Motivation and results      Outline of proof   Ergodic self-similarity

Invariant sets and measures

Deﬁnition
Let X be a space and T : X → X a transformation.
A set E ⊂ X is invariant if TE ⊂ E.
A measure µ on X is invariant if T µ = µ or in other words,
µ(T −1 A) = µ(A) for all measurable sets A.

Example (Morphisms of the circle)
Let S 1 = R/Z = [0, 1] with endpoints identiﬁed. For m ≥ 2 let
Tm : S 1 → S 1 be the “times m” map:

Tm (x) = m · x mod 1.

These are the simplest examples of chaotic transformations.
Background            Motivation and results      Outline of proof   Ergodic self-similarity

Invariant sets and measures

Deﬁnition
Let X be a space and T : X → X a transformation.
A set E ⊂ X is invariant if TE ⊂ E.
A measure µ on X is invariant if T µ = µ or in other words,
µ(T −1 A) = µ(A) for all measurable sets A.

Example (Morphisms of the circle)
Let S 1 = R/Z = [0, 1] with endpoints identiﬁed. For m ≥ 2 let
Tm : S 1 → S 1 be the “times m” map:

Tm (x) = m · x mod 1.

These are the simplest examples of chaotic transformations.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Measures invariant under Tm

There is a breathtaking variety of sets and measures invariant
under Tm . Some examples:
1   All of S 1 (set) and Lebesgue measure.
2   A periodic orbit (set) and the uniform discrete measure on
a periodic orbit (measure).
3   The middle-third Cantor set is invariant under T3 . Any
Bernoulli measure on it is also invariant.
4   These are just the simplest ones, there are many more,
including many with “pathological” properties (e.g. minimal
but positive entropy).

Question
What about sets/measures invariant under both T2 and T3 ?
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Measures invariant under Tm

There is a breathtaking variety of sets and measures invariant
under Tm . Some examples:
1   All of S 1 (set) and Lebesgue measure.
2   A periodic orbit (set) and the uniform discrete measure on
a periodic orbit (measure).
3   The middle-third Cantor set is invariant under T3 . Any
Bernoulli measure on it is also invariant.
4   These are just the simplest ones, there are many more,
including many with “pathological” properties (e.g. minimal
but positive entropy).

Question
What about sets/measures invariant under both T2 and T3 ?
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Measures invariant under Tm

There is a breathtaking variety of sets and measures invariant
under Tm . Some examples:
1   All of S 1 (set) and Lebesgue measure.
2   A periodic orbit (set) and the uniform discrete measure on
a periodic orbit (measure).
3   The middle-third Cantor set is invariant under T3 . Any
Bernoulli measure on it is also invariant.
4   These are just the simplest ones, there are many more,
including many with “pathological” properties (e.g. minimal
but positive entropy).

Question
What about sets/measures invariant under both T2 and T3 ?
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Measures invariant under Tm

There is a breathtaking variety of sets and measures invariant
under Tm . Some examples:
1   All of S 1 (set) and Lebesgue measure.
2   A periodic orbit (set) and the uniform discrete measure on
a periodic orbit (measure).
3   The middle-third Cantor set is invariant under T3 . Any
Bernoulli measure on it is also invariant.
4   These are just the simplest ones, there are many more,
including many with “pathological” properties (e.g. minimal
but positive entropy).

Question
What about sets/measures invariant under both T2 and T3 ?
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Measures invariant under Tm

There is a breathtaking variety of sets and measures invariant
under Tm . Some examples:
1   All of S 1 (set) and Lebesgue measure.
2   A periodic orbit (set) and the uniform discrete measure on
a periodic orbit (measure).
3   The middle-third Cantor set is invariant under T3 . Any
Bernoulli measure on it is also invariant.
4   These are just the simplest ones, there are many more,
including many with “pathological” properties (e.g. minimal
but positive entropy).

Question
What about sets/measures invariant under both T2 and T3 ?
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Measures invariant under Tm

There is a breathtaking variety of sets and measures invariant
under Tm . Some examples:
1   All of S 1 (set) and Lebesgue measure.
2   A periodic orbit (set) and the uniform discrete measure on
a periodic orbit (measure).
3   The middle-third Cantor set is invariant under T3 . Any
Bernoulli measure on it is also invariant.
4   These are just the simplest ones, there are many more,
including many with “pathological” properties (e.g. minimal
but positive entropy).

Question
What about sets/measures invariant under both T2 and T3 ?
Background          Motivation and results   Outline of proof   Ergodic self-similarity

Furstenberg’s principle

Heuristic principle
The dynamics of T2 and T3 are independent.
Another way to put it: expansions in base 2 and base 3 have
nothing to do with each other.

Theorem (Furstenberg, 1967)
If E ⊂ S 1 is closed and invariant under T2 and T3 , then E is
either ﬁnite or S 1 .

Big Open Problem
Let µ be a measure invariant under T2 and T3 . Is it true that µ
has to be a linear combination of measures supported on ﬁnite
orbits and Lebesgue?
Background          Motivation and results   Outline of proof   Ergodic self-similarity

Furstenberg’s principle

Heuristic principle
The dynamics of T2 and T3 are independent.
Another way to put it: expansions in base 2 and base 3 have
nothing to do with each other.

Theorem (Furstenberg, 1967)
If E ⊂ S 1 is closed and invariant under T2 and T3 , then E is
either ﬁnite or S 1 .

Big Open Problem
Let µ be a measure invariant under T2 and T3 . Is it true that µ
has to be a linear combination of measures supported on ﬁnite
orbits and Lebesgue?
Background          Motivation and results   Outline of proof   Ergodic self-similarity

Furstenberg’s principle

Heuristic principle
The dynamics of T2 and T3 are independent.
Another way to put it: expansions in base 2 and base 3 have
nothing to do with each other.

Theorem (Furstenberg, 1967)
If E ⊂ S 1 is closed and invariant under T2 and T3 , then E is
either ﬁnite or S 1 .

Big Open Problem
Let µ be a measure invariant under T2 and T3 . Is it true that µ
has to be a linear combination of measures supported on ﬁnite
orbits and Lebesgue?
Background         Motivation and results   Outline of proof   Ergodic self-similarity

Rudolph’s Theorem

Theorem (D. Rudolph, 1990)
If µ is invariant under T2 and T3 and dim∗ µ > 0, then µ is
Lebesgue measure.
Background             Motivation and results        Outline of proof      Ergodic self-similarity

Convolutions of measures

Deﬁnition
If µ, ν are measures on R, their convolution is

(µ ∗ ν)(A) =             µ(x − A)dν(x) =          ν(x − A)dµ(x).

Remark
Alternatively,
µ ∗ ν = S(µ × ν),
where S(x, y ) = x + y is the addition map. This is essentially
an orthogonal projection (along a 45◦ line). This provides a link
with the projection problem.
Background             Motivation and results        Outline of proof      Ergodic self-similarity

Convolutions of measures

Deﬁnition
If µ, ν are measures on R, their convolution is

(µ ∗ ν)(A) =             µ(x − A)dν(x) =          ν(x − A)dµ(x).

Remark
Alternatively,
µ ∗ ν = S(µ × ν),
where S(x, y ) = x + y is the addition map. This is essentially
an orthogonal projection (along a 45◦ line). This provides a link
with the projection problem.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

A conjecture of Furstenberg

Conjecture (H. Furstenberg)
Let µ be invariant under T2 and ν be invariant under T3 . Then

dim∗ (µ ∗ ν) = min(dim∗ µ + dim∗ ν, 1).

Furstenberg stated the conjecture in terms of sets, not
measures. The measure version is stronger.
This says that µ and ν cannot resonate at arbitrarily small
scales.
This conjecture implies Rudolph’s Theorem.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

A conjecture of Furstenberg

Conjecture (H. Furstenberg)
Let µ be invariant under T2 and ν be invariant under T3 . Then

dim∗ (µ ∗ ν) = min(dim∗ µ + dim∗ ν, 1).

Furstenberg stated the conjecture in terms of sets, not
measures. The measure version is stronger.
This says that µ and ν cannot resonate at arbitrarily small
scales.
This conjecture implies Rudolph’s Theorem.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

A conjecture of Furstenberg

Conjecture (H. Furstenberg)
Let µ be invariant under T2 and ν be invariant under T3 . Then

dim∗ (µ ∗ ν) = min(dim∗ µ + dim∗ ν, 1).

Furstenberg stated the conjecture in terms of sets, not
measures. The measure version is stronger.
This says that µ and ν cannot resonate at arbitrarily small
scales.
This conjecture implies Rudolph’s Theorem.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

A conjecture of Furstenberg

Conjecture (H. Furstenberg)
Let µ be invariant under T2 and ν be invariant under T3 . Then

dim∗ (µ ∗ ν) = min(dim∗ µ + dim∗ ν, 1).

Furstenberg stated the conjecture in terms of sets, not
measures. The measure version is stronger.
This says that µ and ν cannot resonate at arbitrarily small
scales.
This conjecture implies Rudolph’s Theorem.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Main result

Theorem (M. Hochman and P.S., 2009)
Furstenberg’s conjecture is true.

Remarks
The key element in the proof is that, for certain measures
which satisfy an ergodic-theoretic form of self-similarity, the
dimension of projections behaves in a semicontinuous way.
This general result yields many other concrete examples
where one can ﬁnd the precise set of exceptions in the
projection theorem, in particular recovering, unifying and
extending previous results by Moreira, Peres-S. and
Nazarov-Peres-S.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Main result

Theorem (M. Hochman and P.S., 2009)
Furstenberg’s conjecture is true.

Remarks
The key element in the proof is that, for certain measures
which satisfy an ergodic-theoretic form of self-similarity, the
dimension of projections behaves in a semicontinuous way.
This general result yields many other concrete examples
where one can ﬁnd the precise set of exceptions in the
projection theorem, in particular recovering, unifying and
extending previous results by Moreira, Peres-S. and
Nazarov-Peres-S.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Main result

Theorem (M. Hochman and P.S., 2009)
Furstenberg’s conjecture is true.

Remarks
The key element in the proof is that, for certain measures
which satisfy an ergodic-theoretic form of self-similarity, the
dimension of projections behaves in a semicontinuous way.
This general result yields many other concrete examples
where one can ﬁnd the precise set of exceptions in the
projection theorem, in particular recovering, unifying and
extending previous results by Moreira, Peres-S. and
Nazarov-Peres-S.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Main result

Theorem (M. Hochman and P.S., 2009)
Furstenberg’s conjecture is true.

Remarks
The key element in the proof is that, for certain measures
which satisfy an ergodic-theoretic form of self-similarity, the
dimension of projections behaves in a semicontinuous way.
This general result yields many other concrete examples
where one can ﬁnd the precise set of exceptions in the
projection theorem, in particular recovering, unifying and
extending previous results by Moreira, Peres-S. and
Nazarov-Peres-S.
Background            Motivation and results    Outline of proof     Ergodic self-similarity

Outline of proof

Let St (x, y ) = x + ty . It follows from (a reparametrization
of) the projection theorem that
dim∗ (St (µ × ν)) = min(dim∗ µ + dim∗ ν, 1)
for almost every t.
By our general semicontinuity results (and additional
work!), for any ε > 0, the set
Uε := {t : dim∗ (St (µ × ν)) > min(dim∗ µ + dim∗ ν, 1) − ε}
is open (and dense).
Using that µ and ν are invariant under T2 and T3 , one
shows that Uε is invariant under multiplication by 2 and
1/3.
Since log 3/ log 2 is irrational, it follows that Uε = R \ {0}.
Background            Motivation and results    Outline of proof     Ergodic self-similarity

Outline of proof

Let St (x, y ) = x + ty . It follows from (a reparametrization
of) the projection theorem that
dim∗ (St (µ × ν)) = min(dim∗ µ + dim∗ ν, 1)
for almost every t.
By our general semicontinuity results (and additional
work!), for any ε > 0, the set
Uε := {t : dim∗ (St (µ × ν)) > min(dim∗ µ + dim∗ ν, 1) − ε}
is open (and dense).
Using that µ and ν are invariant under T2 and T3 , one
shows that Uε is invariant under multiplication by 2 and
1/3.
Since log 3/ log 2 is irrational, it follows that Uε = R \ {0}.
Background            Motivation and results    Outline of proof     Ergodic self-similarity

Outline of proof

Let St (x, y ) = x + ty . It follows from (a reparametrization
of) the projection theorem that
dim∗ (St (µ × ν)) = min(dim∗ µ + dim∗ ν, 1)
for almost every t.
By our general semicontinuity results (and additional
work!), for any ε > 0, the set
Uε := {t : dim∗ (St (µ × ν)) > min(dim∗ µ + dim∗ ν, 1) − ε}
is open (and dense).
Using that µ and ν are invariant under T2 and T3 , one
shows that Uε is invariant under multiplication by 2 and
1/3.
Since log 3/ log 2 is irrational, it follows that Uε = R \ {0}.
Background            Motivation and results    Outline of proof     Ergodic self-similarity

Outline of proof

Let St (x, y ) = x + ty . It follows from (a reparametrization
of) the projection theorem that
dim∗ (St (µ × ν)) = min(dim∗ µ + dim∗ ν, 1)
for almost every t.
By our general semicontinuity results (and additional
work!), for any ε > 0, the set
Uε := {t : dim∗ (St (µ × ν)) > min(dim∗ µ + dim∗ ν, 1) − ε}
is open (and dense).
Using that µ and ν are invariant under T2 and T3 , one
shows that Uε is invariant under multiplication by 2 and
1/3.
Since log 3/ log 2 is irrational, it follows that Uε = R \ {0}.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

Zooming-in sequences

Let µ be a measure on [0, 1]2 and ﬁx a base p ≥ 2.
Given x ∈ suppµ, we deﬁne a sequence of measures µn,x
corresponding to zooming in p-adically towards x:

Let Qn be the p-adic square of side length p−n containing
x.
Let µn,x be the normalized restriction of µ to Qn .
Finally, we obtain µn,x by rescaling µn,x back to the unit
square.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

Zooming-in sequences

Let µ be a measure on [0, 1]2 and ﬁx a base p ≥ 2.
Given x ∈ suppµ, we deﬁne a sequence of measures µn,x
corresponding to zooming in p-adically towards x:

Let Qn be the p-adic square of side length p−n containing
x.
Let µn,x be the normalized restriction of µ to Qn .
Finally, we obtain µn,x by rescaling µn,x back to the unit
square.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

Zooming-in sequences

Let µ be a measure on [0, 1]2 and ﬁx a base p ≥ 2.
Given x ∈ suppµ, we deﬁne a sequence of measures µn,x
corresponding to zooming in p-adically towards x:

Let Qn be the p-adic square of side length p−n containing
x.
Let µn,x be the normalized restriction of µ to Qn .
Finally, we obtain µn,x by rescaling µn,x back to the unit
square.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

Zooming-in sequences

Let µ be a measure on [0, 1]2 and ﬁx a base p ≥ 2.
Given x ∈ suppµ, we deﬁne a sequence of measures µn,x
corresponding to zooming in p-adically towards x:

Let Qn be the p-adic square of side length p−n containing
x.
Let µn,x be the normalized restriction of µ to Qn .
Finally, we obtain µn,x by rescaling µn,x back to the unit
square.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

Zooming-in sequences

Let µ be a measure on [0, 1]2 and ﬁx a base p ≥ 2.
Given x ∈ suppµ, we deﬁne a sequence of measures µn,x
corresponding to zooming in p-adically towards x:

Let Qn be the p-adic square of side length p−n containing
x.
Let µn,x be the normalized restriction of µ to Qn .
Finally, we obtain µn,x by rescaling µn,x back to the unit
square.
Background         Motivation and results                 Outline of proof   Ergodic self-similarity

Local dimension and entropy averages

Fix a base p ≥ 2, and let

H(µ) =                 −µ(Q) log(µ(Q)),
Q

where the sum ranges over all p-adic squares of ﬁrst level.
Lemma (Local entropy averages)
Let µ be any measure on [0, 1]2 . Then for µ-almost every x,
N
1
lim inf                          H(µn,x ) = dim(µ, x).
N→∞       N log p
n=1

In other words, local dimension can be estimated through local
entropy averages at a ﬁxed scale 1/p.
Background         Motivation and results                 Outline of proof   Ergodic self-similarity

Local dimension and entropy averages

Fix a base p ≥ 2, and let

H(µ) =                 −µ(Q) log(µ(Q)),
Q

where the sum ranges over all p-adic squares of ﬁrst level.
Lemma (Local entropy averages)
Let µ be any measure on [0, 1]2 . Then for µ-almost every x,
N
1
lim inf                          H(µn,x ) = dim(µ, x).
N→∞       N log p
n=1

In other words, local dimension can be estimated through local
entropy averages at a ﬁxed scale 1/p.
Background         Motivation and results                 Outline of proof   Ergodic self-similarity

Local dimension and entropy averages

Fix a base p ≥ 2, and let

H(µ) =                 −µ(Q) log(µ(Q)),
Q

where the sum ranges over all p-adic squares of ﬁrst level.
Lemma (Local entropy averages)
Let µ be any measure on [0, 1]2 . Then for µ-almost every x,
N
1
lim inf                          H(µn,x ) = dim(µ, x).
N→∞       N log p
n=1

In other words, local dimension can be estimated through local
entropy averages at a ﬁxed scale 1/p.
Background         Motivation and results                 Outline of proof   Ergodic self-similarity

Local dimension and entropy averages

Fix a base p ≥ 2, and let

H(µ) =                 −µ(Q) log(µ(Q)),
Q

where the sum ranges over all p-adic squares of ﬁrst level.
Lemma (Local entropy averages)
Let µ be any measure on [0, 1]2 . Then for µ-almost every x,
N
1
lim inf                          H(µn,x ) = dim(µ, x).
N→∞       N log p
n=1

In other words, local dimension can be estimated through local
entropy averages at a ﬁxed scale 1/p.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Remarks on the entropy averages lemma

The proof is a very simple martingale argument due to Y.
Peres.
Similar ideas have been used, result itself might not be
new.
Usually dimension is calculated either locally or globally.
The entropy average approach is semi-local: one zooms-in
towards a point but at each scale one looks at the
distribution of the measure in a neighborhood.
This works especially well when passing to projections,
thanks to convexity of the entropy function.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Remarks on the entropy averages lemma

The proof is a very simple martingale argument due to Y.
Peres.
Similar ideas have been used, result itself might not be
new.
Usually dimension is calculated either locally or globally.
The entropy average approach is semi-local: one zooms-in
towards a point but at each scale one looks at the
distribution of the measure in a neighborhood.
This works especially well when passing to projections,
thanks to convexity of the entropy function.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Remarks on the entropy averages lemma

The proof is a very simple martingale argument due to Y.
Peres.
Similar ideas have been used, result itself might not be
new.
Usually dimension is calculated either locally or globally.
The entropy average approach is semi-local: one zooms-in
towards a point but at each scale one looks at the
distribution of the measure in a neighborhood.
This works especially well when passing to projections,
thanks to convexity of the entropy function.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Remarks on the entropy averages lemma

The proof is a very simple martingale argument due to Y.
Peres.
Similar ideas have been used, result itself might not be
new.
Usually dimension is calculated either locally or globally.
The entropy average approach is semi-local: one zooms-in
towards a point but at each scale one looks at the
distribution of the measure in a neighborhood.
This works especially well when passing to projections,
thanks to convexity of the entropy function.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

The idea of ergodic self-similarity

Usual self-similarity applies to a single set/measure.
Ergodic self-similarity is a property not of a single
measure, but of a probability distribution P on measures.
P is self-similar in the sense that if µ is selected according
to P and one zooms in towards a random point x (selected
according to µ), we obtain again a random measure with
distribution P.
This is related to, but vastly more general than, stochastic
self-similarity.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

The idea of ergodic self-similarity

Usual self-similarity applies to a single set/measure.
Ergodic self-similarity is a property not of a single
measure, but of a probability distribution P on measures.
P is self-similar in the sense that if µ is selected according
to P and one zooms in towards a random point x (selected
according to µ), we obtain again a random measure with
distribution P.
This is related to, but vastly more general than, stochastic
self-similarity.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

The idea of ergodic self-similarity

Usual self-similarity applies to a single set/measure.
Ergodic self-similarity is a property not of a single
measure, but of a probability distribution P on measures.
P is self-similar in the sense that if µ is selected according
to P and one zooms in towards a random point x (selected
according to µ), we obtain again a random measure with
distribution P.
This is related to, but vastly more general than, stochastic
self-similarity.
Background            Motivation and results   Outline of proof    Ergodic self-similarity

The idea of ergodic self-similarity

Usual self-similarity applies to a single set/measure.
Ergodic self-similarity is a property not of a single
measure, but of a probability distribution P on measures.
P is self-similar in the sense that if µ is selected according
to P and one zooms in towards a random point x (selected
according to µ), we obtain again a random measure with
distribution P.
This is related to, but vastly more general than, stochastic
self-similarity.
Background            Motivation and results           Outline of proof   Ergodic self-similarity

CP Chains

Let M denote the set of all probability measures on [0, 1]2 .
Fix a base p ≥ 2. Let Q denote the family of p2 p-adic
squares of ﬁrst level.
If Q ⊂ [0, 1]2 is any square and M we denote by µQ the
measure obtained by “zooming in” into Q. More precisely,
µQ is obtained by restricting µ to Q, normalizing, and
mapping back homothetically to the unit square.
Given µ ∈ M, the transition law is

µ → µQ          with probability µ(Q).
Background            Motivation and results           Outline of proof   Ergodic self-similarity

CP Chains

Let M denote the set of all probability measures on [0, 1]2 .
Fix a base p ≥ 2. Let Q denote the family of p2 p-adic
squares of ﬁrst level.
If Q ⊂ [0, 1]2 is any square and M we denote by µQ the
measure obtained by “zooming in” into Q. More precisely,
µQ is obtained by restricting µ to Q, normalizing, and
mapping back homothetically to the unit square.
Given µ ∈ M, the transition law is

µ → µQ          with probability µ(Q).
Background            Motivation and results           Outline of proof   Ergodic self-similarity

CP Chains

Let M denote the set of all probability measures on [0, 1]2 .
Fix a base p ≥ 2. Let Q denote the family of p2 p-adic
squares of ﬁrst level.
If Q ⊂ [0, 1]2 is any square and M we denote by µQ the
measure obtained by “zooming in” into Q. More precisely,
µQ is obtained by restricting µ to Q, normalizing, and
mapping back homothetically to the unit square.
Given µ ∈ M, the transition law is

µ → µQ          with probability µ(Q).
Background            Motivation and results           Outline of proof   Ergodic self-similarity

CP Chains

Let M denote the set of all probability measures on [0, 1]2 .
Fix a base p ≥ 2. Let Q denote the family of p2 p-adic
squares of ﬁrst level.
If Q ⊂ [0, 1]2 is any square and M we denote by µQ the
measure obtained by “zooming in” into Q. More precisely,
µQ is obtained by restricting µ to Q, normalizing, and
mapping back homothetically to the unit square.
Given µ ∈ M, the transition law is

µ → µQ          with probability µ(Q).
Background             Motivation and results    Outline of proof      Ergodic self-similarity

Ergodic fractal measures

Let P be a measure on M (a measure on measures!).
Recall that P is stationary for the Markov chain if,
whenever µ is selected randomly according to P and then
the transition applied, the resulting measure is again
distributed according to P.
If µ1 is drawn randomly according to P and we obtain a
sequence µ1 , µ2 , µ3 , . . . by following the transition law, each
µi is also distributed according to P.
Typical measures for P will be denoted (following
Furstenberg) ergodic fractal measures.
P may or may not be ergodic.
Background             Motivation and results    Outline of proof      Ergodic self-similarity

Ergodic fractal measures

Let P be a measure on M (a measure on measures!).
Recall that P is stationary for the Markov chain if,
whenever µ is selected randomly according to P and then
the transition applied, the resulting measure is again
distributed according to P.
If µ1 is drawn randomly according to P and we obtain a
sequence µ1 , µ2 , µ3 , . . . by following the transition law, each
µi is also distributed according to P.
Typical measures for P will be denoted (following
Furstenberg) ergodic fractal measures.
P may or may not be ergodic.
Background             Motivation and results    Outline of proof      Ergodic self-similarity

Ergodic fractal measures

Let P be a measure on M (a measure on measures!).
Recall that P is stationary for the Markov chain if,
whenever µ is selected randomly according to P and then
the transition applied, the resulting measure is again
distributed according to P.
If µ1 is drawn randomly according to P and we obtain a
sequence µ1 , µ2 , µ3 , . . . by following the transition law, each
µi is also distributed according to P.
Typical measures for P will be denoted (following
Furstenberg) ergodic fractal measures.
P may or may not be ergodic.
Background             Motivation and results    Outline of proof      Ergodic self-similarity

Ergodic fractal measures

Let P be a measure on M (a measure on measures!).
Recall that P is stationary for the Markov chain if,
whenever µ is selected randomly according to P and then
the transition applied, the resulting measure is again
distributed according to P.
If µ1 is drawn randomly according to P and we obtain a
sequence µ1 , µ2 , µ3 , . . . by following the transition law, each
µi is also distributed according to P.
Typical measures for P will be denoted (following
Furstenberg) ergodic fractal measures.
P may or may not be ergodic.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Examples of EFM’s

The following (individual) measures can be related to
appropriate (possibly generalized) CP-chains:
Self-similar measures (with suitable separation).
Stochastically self-similar measures.
Conformal measures for hyperbolic Julia sets.
Measures invariant under Tm .
Measures on the linear ﬁbers of self-similar sets.
Products of the above.
Background         Motivation and results   Outline of proof   Ergodic self-similarity

Micromeasures and ergodic fractal measures

Deﬁnition
Let µ be a measure on [0, 1]. A micromeasure ν of µ is a weak
limit of measures µQn , where Qn are squares of side length
tending to 0. This is closely related to the familiar concept of
tangent measure.

Theorem (Furstenberg 1970,2008)
For any measure µ there is a distribution P supported on its
derived measures, such that the corresponding Markov process
is ergodic.
Moreover, if ν is a typical measure for the process, then
dim∗ (ν) ≥ dim∗ (µ).
Background         Motivation and results   Outline of proof   Ergodic self-similarity

Micromeasures and ergodic fractal measures

Deﬁnition
Let µ be a measure on [0, 1]. A micromeasure ν of µ is a weak
limit of measures µQn , where Qn are squares of side length
tending to 0. This is closely related to the familiar concept of
tangent measure.

Theorem (Furstenberg 1970,2008)
For any measure µ there is a distribution P supported on its
derived measures, such that the corresponding Markov process
is ergodic.
Moreover, if ν is a typical measure for the process, then
dim∗ (ν) ≥ dim∗ (µ).
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Semicontinuity of dimension for EFM’s

Theorem (M.Hochman - P.S. 2009)
Let µ be an ergodic fractal measure. Then there exists a lower
semicontinuous function

E : {orthogonal projections} → [0, 1],

such that:
E(π) = min(dim∗ µ, 1) for almost all π.
dim∗ (πµ) ≥ E(π).

Remark
In many cases, dim∗ (πµ) = E(π), so that this is directly
semicontinuous.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Semicontinuity of dimension for EFM’s

Theorem (M.Hochman - P.S. 2009)
Let µ be an ergodic fractal measure. Then there exists a lower
semicontinuous function

E : {orthogonal projections} → [0, 1],

such that:
E(π) = min(dim∗ µ, 1) for almost all π.
dim∗ (πµ) ≥ E(π).

Remark
In many cases, dim∗ (πµ) = E(π), so that this is directly
semicontinuous.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Semicontinuity of dimension for EFM’s

Theorem (M.Hochman - P.S. 2009)
Let µ be an ergodic fractal measure. Then there exists a lower
semicontinuous function

E : {orthogonal projections} → [0, 1],

such that:
E(π) = min(dim∗ µ, 1) for almost all π.
dim∗ (πµ) ≥ E(π).

Remark
In many cases, dim∗ (πµ) = E(π), so that this is directly
semicontinuous.
Background            Motivation and results   Outline of proof   Ergodic self-similarity

Semicontinuity of dimension for EFM’s

Theorem (M.Hochman - P.S. 2009)
Let µ be an ergodic fractal measure. Then there exists a lower
semicontinuous function

E : {orthogonal projections} → [0, 1],

such that:
E(π) = min(dim∗ µ, 1) for almost all π.
dim∗ (πµ) ≥ E(π).

Remark
In many cases, dim∗ (πµ) = E(π), so that this is directly
semicontinuous.
Background   Motivation and results   Outline of proof   Ergodic self-similarity

That’s it

The end

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 9 posted: 1/22/2011 language: English pages: 114