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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