Extremality and dynamically defined measures David Simmons Extremality and dynamically defined measures Diophantine preliminaries First results Main results David Simmons Quasi- decaying University of York measures
Extremality and dynamically defined measures 1 Diophantine preliminaries David Simmons Diophantine preliminaries 2 First results First results Main results Quasi- 3 Main results decaying measures 4 Quasi-decaying measures
References Extremality and dynamically defined measures David Simmons T. Das, L. Fishman, D. S. Simmons, and M. Urba´ nski, Extremality and dynamically defined measures, I: Diophantine preliminaries Diophantine properties of quasi-decaying measures , First results http://arxiv.org/abs/1504.04778 , preprint 2015. Main results , Extremality and dynamically defined measures, II: Quasi- decaying Measures from conformal dynamical systems , measures http://arxiv.org/abs/1508.05592 , preprint 2015.
Very well approximable vectors Extremality and Definition dynamically defined A vector x ∈ R d is very well approximable if there exists ε > 0 measures such that for infinitely many p / q ∈ Q d , David Simmons � � � � 1 Diophantine � x − p � � � ≤ q 1+1 / d + ε · preliminaries q First results Main results Quasi- decaying measures
Very well approximable vectors Extremality and Definition dynamically defined A vector x ∈ R d is very well approximable if there exists ε > 0 measures such that for infinitely many p / q ∈ Q d , David Simmons � � � � 1 Diophantine � x − p � � � ≤ q 1+1 / d + ε · preliminaries q First results Main results Quasi- Example decaying measures Roth’s theorem states that no algebraic irrational number in R is very well approximable. Its higher-dimensional generalization (a corollary of Schmidt’s subspace theorem) says that an algebraic vector in R d is very well approximable if and only if it is contained in an affine rational subspace of R d .
Dynamical interpretation Extremality and dynamically defined Theorem (Kleinbock–Margulis ’99) measures David Let Simmons � � � I d � e t / d I d Diophantine − x preliminaries g t = , u x = , e − t 1 First results Λ ∗ = Z d +1 ∈ Ω d +1 = { unimodular lattices in R d +1 } . Main results Quasi- decaying Then x is very well approximable if and only if measures 1 lim sup t dist Ω d +1 (Λ ∗ , g t u x Λ ∗ ) > 0 . t →∞
Extremal measures Extremality and A measure on R d is called extremal if it gives full measure to dynamically defined the set of not very well approximable vectors. measures David Simmons Example (Corollary of Borel–Cantelli) Lebesgue measure on R d is extremal. Diophantine preliminaries First results Main results Quasi- decaying measures
Extremal measures Extremality and A measure on R d is called extremal if it gives full measure to dynamically defined the set of not very well approximable vectors. measures David Simmons Example (Corollary of Borel–Cantelli) Lebesgue measure on R d is extremal. Diophantine preliminaries First results Conjecture (Mahler ’32, proven by Sprindˇ zuk ’64) Main results Quasi- Lebesgue measure on { ( x , x 2 , . . . , x d ) : x ∈ R } is extremal. decaying measures
Extremal measures Extremality and A measure on R d is called extremal if it gives full measure to dynamically defined the set of not very well approximable vectors. measures David Simmons Example (Corollary of Borel–Cantelli) Lebesgue measure on R d is extremal. Diophantine preliminaries First results Conjecture (Mahler ’32, proven by Sprindˇ zuk ’64) Main results Quasi- Lebesgue measure on { ( x , x 2 , . . . , x d ) : x ∈ R } is extremal. decaying measures Conjecture (Sprindˇ zuk ’80, proven by Kleinbock–Margulis ’98) Lebesgue measure on any real-analytic manifold not contained in an affine hyperplane is extremal.
Extremality and dynamically defined measures: First results Extremality and dynamically defined Theorem (Klenbock–Lindenstrauss–Weiss ’04) measures David Simmons Let Λ be the limit set of a finite iterated function system generated by similarities and satisfying the open set condition, Diophantine preliminaries and let δ = dim H (Λ) . Suppose that Λ is not contained in any First results affine hyperplane. Then H δ ↿ Λ is extremal. Main results Quasi- decaying measures
Extremality and dynamically defined measures: First results Extremality and dynamically defined Theorem (Klenbock–Lindenstrauss–Weiss ’04) measures David Simmons Let Λ be the limit set of a finite iterated function system generated by similarities and satisfying the open set condition, Diophantine preliminaries and let δ = dim H (Λ) . Suppose that Λ is not contained in any First results affine hyperplane. Then H δ ↿ Λ is extremal. Main results Quasi- Theorem (Urba´ nski ’05) decaying measures Same is true if “similarities” is replaced by “conformal maps”, and if H δ ↿ Λ is replaced by “the Gibbs measure of a H¨ older continuous potential function”.
Extremality and dynamically defined measures: First results Extremality and dynamically defined Theorem (Stratmann–Urba´ nski ’06) measures David Let G be a convex-cocompact Kleinian group whose limit set is Simmons not contained in any affine hyperplane. Then the Diophantine Patterson–Sullivan measure of G is extremal. preliminaries First results Main results Quasi- decaying measures
Extremality and dynamically defined measures: First results Extremality and dynamically defined Theorem (Stratmann–Urba´ nski ’06) measures David Let G be a convex-cocompact Kleinian group whose limit set is Simmons not contained in any affine hyperplane. Then the Diophantine Patterson–Sullivan measure of G is extremal. preliminaries First results Theorem (Urba´ nski ’05 + Markov partition argument) Main results Quasi- Let T : � C → � C be a hyperbolic (i.e. expansive on its Julia set) decaying measures rational function, let φ : � C → R be a H¨ older continuous potential function, and let µ φ be the corresponding Gibbs measure. If Supp( µ φ ) is not contained in an affine hyperplane, then µ φ is extremal.
Friendly and absolutely friendly measures These theorems in fact all prove a stronger condition than Extremality and extremality, namely friendliness . dynamically defined measures Definition (Kleinbock–Lindenstrauss–Weiss ’04) David Simmons A measure µ is called friendly (resp. absolutely friendly ) if: Diophantine µ is doubling and gives zero measure to every hyperplane. preliminaries There exist C 1 , α > 0 such that for every ball B = B ( x , ρ ) First results Main results with x ∈ Supp( µ ), for every 0 < β ≤ 1, and for every Quasi- hyperplane L ⊆ R d , decaying measures � � ≤ C 1 β α µ ( B ) (decaying) µ N ( L , β ess sup d ( · , L )) ∩ B B resp. � � ≤ C 1 β α µ ( B ) (absolutely decaying) µ N ( L , βρ ) ∩ B
Friendly and absolutely friendly measures Extremality and Theorem (Kleinbock–Lindenstraus–Weiss ’04) dynamically defined Every friendly measure is extremal. measures David Simmons Diophantine preliminaries First results Main results Quasi- decaying measures
Friendly and absolutely friendly measures Extremality and Theorem (Kleinbock–Lindenstraus–Weiss ’04) dynamically defined Every friendly measure is extremal. measures David Simmons Theorem (Kleinbock–Lindenstraus–Weiss ’04) Diophantine If Φ : R k → R d is a real-analytic embedding whose image is not preliminaries First results contained in any affine hyperplane, then Φ sends absolutely Main results friendly measures to friendly measures. Quasi- decaying measures
Friendly and absolutely friendly measures Extremality and Theorem (Kleinbock–Lindenstraus–Weiss ’04) dynamically defined Every friendly measure is extremal. measures David Simmons Theorem (Kleinbock–Lindenstraus–Weiss ’04) Diophantine If Φ : R k → R d is a real-analytic embedding whose image is not preliminaries First results contained in any affine hyperplane, then Φ sends absolutely Main results friendly measures to friendly measures. Quasi- decaying measures Theorem (Folklore) If δ > d − 1 , then every Ahlfors δ -regular measure on R d is absolutely friendly.
Friendly and absolutely friendly measures Extremality and Theorem (Kleinbock–Lindenstraus–Weiss ’04) dynamically defined Every friendly measure is extremal. measures David Simmons Theorem (Kleinbock–Lindenstraus–Weiss ’04) Diophantine If Φ : R k → R d is a real-analytic embedding whose image is not preliminaries First results contained in any affine hyperplane, then Φ sends absolutely Main results friendly measures to friendly measures. Quasi- decaying measures Theorem (Folklore) If δ > d − 1 , then every Ahlfors δ -regular measure on R d is absolutely friendly. Philosophical meta-theorem: Every Ahlfors regular “nonplanar” measure is absolutely friendly.
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