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?

      Answer (One of many possibilities)
      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?

      Answer (One of many possibilities)
      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?

      Answer (One of many possibilities)
      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?

      Answer (One of many possibilities)
      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.
             What about a typical projection?
             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.
             What about a typical projection?
             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.
             What about a typical projection?
             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.
             What about a typical projection?
             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.
             What about a typical projection?
             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.
             Furstenberg about independent dynamics.
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.
             Furstenberg about independent dynamics.
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.
             Furstenberg about independent dynamics.
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.
             Furstenberg about independent dynamics.
Background            Motivation and results   Outline of proof   Ergodic self-similarity



Hillel Furstenberg


             American-Israeli mathematician. Made fundamental
             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


             American-Israeli mathematician. Made fundamental
             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


             American-Israeli mathematician. Made fundamental
             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


             American-Israeli mathematician. Made fundamental
             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


             American-Israeli mathematician. Made fundamental
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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.
             Gibbs measures on cookie-cutters.
             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

				
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