Additive Combinatorics methods in Fractal Geometry II Pablo Shmerkin Department of Mathematics and Statistics T. Di Tella University and CONICET Dynamics Beyond Uniform Hyperbolicity, CIRM, May 2019 P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 1 / 25
Review: dynamical self-similarity Definition G is a compact Abelian group, and h ∈ G is such that the orbit { nh : n ∈ Z } is dense. We let T ( g ) = g + h . λ ∈ ( 0 , 1 ) is a contraction parameter. ∆( x ) : G → A d C is a map taking values in purely atomic measures in R d with at most C atoms. We call ( G , T , λ, ∆) a model. The measures µ x = ∗ ∞ n = 1 S λ n ∆( T n x ) are called dynamical self-similar measures generated by the model. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 2 / 25
Review: dynamical self-similarity Definition G is a compact Abelian group, and h ∈ G is such that the orbit { nh : n ∈ Z } is dense. We let T ( g ) = g + h . λ ∈ ( 0 , 1 ) is a contraction parameter. ∆( x ) : G → A d C is a map taking values in purely atomic measures in R d with at most C atoms. We call ( G , T , λ, ∆) a model. The measures µ x = ∗ ∞ n = 1 S λ n ∆( T n x ) are called dynamical self-similar measures generated by the model. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 2 / 25
Review: dynamical self-similarity Definition G is a compact Abelian group, and h ∈ G is such that the orbit { nh : n ∈ Z } is dense. We let T ( g ) = g + h . λ ∈ ( 0 , 1 ) is a contraction parameter. ∆( x ) : G → A d C is a map taking values in purely atomic measures in R d with at most C atoms. We call ( G , T , λ, ∆) a model. The measures µ x = ∗ ∞ n = 1 S λ n ∆( T n x ) are called dynamical self-similar measures generated by the model. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 2 / 25
Review: dynamical self-similarity Definition G is a compact Abelian group, and h ∈ G is such that the orbit { nh : n ∈ Z } is dense. We let T ( g ) = g + h . λ ∈ ( 0 , 1 ) is a contraction parameter. ∆( x ) : G → A d C is a map taking values in purely atomic measures in R d with at most C atoms. We call ( G , T , λ, ∆) a model. The measures µ x = ∗ ∞ n = 1 S λ n ∆( T n x ) are called dynamical self-similar measures generated by the model. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 2 / 25
Review: dynamical self-similarity Definition G is a compact Abelian group, and h ∈ G is such that the orbit { nh : n ∈ Z } is dense. We let T ( g ) = g + h . λ ∈ ( 0 , 1 ) is a contraction parameter. ∆( x ) : G → A d C is a map taking values in purely atomic measures in R d with at most C atoms. We call ( G , T , λ, ∆) a model. The measures µ x = ∗ ∞ n = 1 S λ n ∆( T n x ) are called dynamical self-similar measures generated by the model. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 2 / 25
Review: dynamical self-similarity Definition G is a compact Abelian group, and h ∈ G is such that the orbit { nh : n ∈ Z } is dense. We let T ( g ) = g + h . λ ∈ ( 0 , 1 ) is a contraction parameter. ∆( x ) : G → A d C is a map taking values in purely atomic measures in R d with at most C atoms. We call ( G , T , λ, ∆) a model. The measures µ x = ∗ ∞ n = 1 S λ n ∆( T n x ) are called dynamical self-similar measures generated by the model. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 2 / 25
Further examples of dynamical self-similar measures Self-homothetic measures on R d : they correspond to G = { 0 } , λ 1 the (common) contraction of maps in the IFS and ∆ = � i p i δ t i (where t i ∈ R d are translations) is built from the translations and the probabilities of the IFS. If µ, ν are two measures as above with contractions λ 1 , λ 2 , then 2 µ ∗ S e x ν are DSSM where G is a finite group if log λ 2 / log λ 1 ∈ Q , and the circle otherwise. This extends to µ 1 ∗ S e x 2 µ 2 ∗ · · · ∗ S e xm µ m . A homogeneous self-similar measure in dimension d is, by 3 definition, a measure of the form µ = ∗ ∞ n = 1 S λ n O n ∆ , O ∈ O d , λ ∈ ( 0 , 1 ) , ∆ ∈ A . It can be realized as a DSSM where G = � O � , h = O , ∆( g ) = g ∆ . P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 3 / 25
Further examples of dynamical self-similar measures Self-homothetic measures on R d : they correspond to G = { 0 } , λ 1 the (common) contraction of maps in the IFS and ∆ = � i p i δ t i (where t i ∈ R d are translations) is built from the translations and the probabilities of the IFS. If µ, ν are two measures as above with contractions λ 1 , λ 2 , then 2 µ ∗ S e x ν are DSSM where G is a finite group if log λ 2 / log λ 1 ∈ Q , and the circle otherwise. This extends to µ 1 ∗ S e x 2 µ 2 ∗ · · · ∗ S e xm µ m . A homogeneous self-similar measure in dimension d is, by 3 definition, a measure of the form µ = ∗ ∞ n = 1 S λ n O n ∆ , O ∈ O d , λ ∈ ( 0 , 1 ) , ∆ ∈ A . It can be realized as a DSSM where G = � O � , h = O , ∆( g ) = g ∆ . P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 3 / 25
Further examples of dynamical self-similar measures Self-homothetic measures on R d : they correspond to G = { 0 } , λ 1 the (common) contraction of maps in the IFS and ∆ = � i p i δ t i (where t i ∈ R d are translations) is built from the translations and the probabilities of the IFS. If µ, ν are two measures as above with contractions λ 1 , λ 2 , then 2 µ ∗ S e x ν are DSSM where G is a finite group if log λ 2 / log λ 1 ∈ Q , and the circle otherwise. This extends to µ 1 ∗ S e x 2 µ 2 ∗ · · · ∗ S e xm µ m . A homogeneous self-similar measure in dimension d is, by 3 definition, a measure of the form µ = ∗ ∞ n = 1 S λ n O n ∆ , O ∈ O d , λ ∈ ( 0 , 1 ) , ∆ ∈ A . It can be realized as a DSSM where G = � O � , h = O , ∆( g ) = g ∆ . P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 3 / 25
Discrete approximations and shifted self-similarity µ x = ∗ ∞ n = 0 S λ n ∆( T n x ) . We define the discrete step- n approximations µ x , n = ∗ n − 1 j = 0 S λ n ∆( T n x ) . Note that µ x , n is purely atomic with ≤ � n − 1 j = 0 | supp (∆( T j x )) | ≤ C n atoms. We then have the following crucial shifted self-similarity relationship: µ x = µ x , n ∗ S λ n µ T n x . This says that µ x is a convex combination of scaled down (by a factor λ n ) translated copies of µ T n x . P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 4 / 25
Review: Frostman exponent Definition Let µ be a measure on R d . The Frostman exponent dim ∞ ( µ ) is the supremum of all s such that µ ( B ( x , r )) ≤ C s r s for all r ∈ ( 0 , 1 ] and all x . P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 5 / 25
Review: exponential separation Definition We say that a model ( G , T , λ, ∆) has exponential separation if for Haar- almost all x ∈ G there is R > 0 such that following holds for infinitely many n : the atoms of the discrete approximation µ n , x = ∗ n j = 1 S λ j ∆( T j x ) are (distinct and) e − Rn -separated. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 6 / 25
Main Theorem: Frostman exponents of DSSM Theorem (P .S.) Let ( G , T , λ, ∆) be a model with exponential separation on R . We also assume that the maps x �→ ∆( x ) and x �→ µ x are continuous a.e., and that µ x is supported on [ 0 , 1 ] . Let �� � log � ∆( x ) � ∞ dx s = min , 1 , log λ where � ∆ � ∞ = max y ∆ { y } . Then dim ∞ ( µ x ) = s for every x ∈ G. Moreover, for every ε > 0 there is a constant C ε such that, for all x ∈ G, y ∈ R and r ∈ ( 0 , 1 ] , µ x ( B ( y , r )) ≤ C ε r s − ε P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 7 / 25
Remarks on main theorem Remarks In the self-similar case (corresponding to constant ∆ ) a version of this result was obtained by M. Hochman but his version is for Hausdorff dimension rather than Frostman exponents. While exponential separation has to be checked for almost all x, the conclusion holds for all x The transitive translation on a compact Abelian group can be replaced by a uniquely ergodic transformation on a compact metric space. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 8 / 25
Remarks on main theorem Remarks In the self-similar case (corresponding to constant ∆ ) a version of this result was obtained by M. Hochman but his version is for Hausdorff dimension rather than Frostman exponents. While exponential separation has to be checked for almost all x, the conclusion holds for all x The transitive translation on a compact Abelian group can be replaced by a uniquely ergodic transformation on a compact metric space. P . Shmerkin (U.T. Di Tella/CONICET) Additive Combinatorics & Fractals CIRM-Luminy, 14.05.2019 8 / 25
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