Distribution of dense lattice orbits on homogeneous spaces June 6, 2013 Ergodic Theory with connections to arithmetic University of Crete, Iraklion Amos Nevo, Technion based on joint work with Alex Gorodnik Wrawick, ETDS 30
Plan Classical ratio ergodic theorem 1 Kazhdan’s problem : groups of isometries 2 Arnold’s problem : homogeneous spaces 3 Equidistribution of matrices with entries in algebraic number field 4 General duality principle 5 Wrawick, ETDS 30
The classical ratio ergodic theorem • Birkhoff’s pointwise ergodic theorem was generalized to any non-singular Z -action (Hopf [1937], Hurewicz [1944], Chacon-Ornstein [1960]). Let us formulate one important special case. Wrawick, ETDS 30
The classical ratio ergodic theorem • Birkhoff’s pointwise ergodic theorem was generalized to any non-singular Z -action (Hopf [1937], Hurewicz [1944], Chacon-Ornstein [1960]). Let us formulate one important special case. • Assume that T : ( X , λ ) → ( X , λ ) preserves the measure and is conservative and ergodic. Then the ratio theorem states that for u , v ∈ L 1 ( X , λ ) with � v d λ � = 0, the ratios � n k = 0 T k u R n [ u , v ] := � n k = 0 T k v � u d λ converge almost everywhere as n → ∞ to the constant � v d λ Wrawick, ETDS 30
Absence of rate of convergence • Initially, one might expect that it is possible to pick normalization constants a ( n ) satisfying a ( n ) → 0, such that the averages n � n 1 k = 0 u ( T k x ) will converge to a nontrivial limit. a ( n ) Wrawick, ETDS 30
Absence of rate of convergence • Initially, one might expect that it is possible to pick normalization constants a ( n ) satisfying a ( n ) → 0, such that the averages n � n 1 k = 0 u ( T k x ) will converge to a nontrivial limit. a ( n ) • But in fact, J. Aaronson showed that if the action is ergodic and conservative, then for any normalization constants a ( n ) , either for every nonnegative u ∈ L 1 ( X ) , u � = 0 n 1 � u ( T k x ) = 0 lim inf almost everywhere, a ( n ) n →∞ k = 0 Wrawick, ETDS 30
Absence of rate of convergence • Initially, one might expect that it is possible to pick normalization constants a ( n ) satisfying a ( n ) → 0, such that the averages n � n 1 k = 0 u ( T k x ) will converge to a nontrivial limit. a ( n ) • But in fact, J. Aaronson showed that if the action is ergodic and conservative, then for any normalization constants a ( n ) , either for every nonnegative u ∈ L 1 ( X ) , u � = 0 n 1 � u ( T k x ) = 0 lim inf almost everywhere, a ( n ) n →∞ k = 0 • or there exists a subsequence n m such that for nonnegative u ∈ L 1 ( X ) , u � = 0, n m 1 � φ ( T k x ) = ∞ lim almost everywhere. a ( n m ) m →∞ k = 0 Wrawick, ETDS 30
The ratio ergodic theorem for commuting transformations • The generalization of the ratio ergodic theorem to actions of two or more commuting transformations has proved to be a difficult challenge. It was resolved in final form only recently by Hochman [2009], who obtained results directly generalizing the case of general non-singular Z -actions. Important partial results were obtained earlier by Feldman in 2007. Wrawick, ETDS 30
The ratio ergodic theorem for commuting transformations • The generalization of the ratio ergodic theorem to actions of two or more commuting transformations has proved to be a difficult challenge. It was resolved in final form only recently by Hochman [2009], who obtained results directly generalizing the case of general non-singular Z -actions. Important partial results were obtained earlier by Feldman in 2007. • As always, one has to decide on a sequence of asymptotically invariant sets in Z d to sum over. The crucial property the (symmetric) family must satisfy was shown by Hochman to be the Besicovich covering property. There are counterexamples to the ratio ergodic theorem when summing over general asymptotically invariant sequences due to Brunel and Krengel [1985]. Wrawick, ETDS 30
The ratio ergodic theorem for commuting transformations • The generalization of the ratio ergodic theorem to actions of two or more commuting transformations has proved to be a difficult challenge. It was resolved in final form only recently by Hochman [2009], who obtained results directly generalizing the case of general non-singular Z -actions. Important partial results were obtained earlier by Feldman in 2007. • As always, one has to decide on a sequence of asymptotically invariant sets in Z d to sum over. The crucial property the (symmetric) family must satisfy was shown by Hochman to be the Besicovich covering property. There are counterexamples to the ratio ergodic theorem when summing over general asymptotically invariant sequences due to Brunel and Krengel [1985]. • As to non-amenable groups, let us now describe some recent examples of general ratio ergodic theorems and also of some counterexamples. Wrawick, ETDS 30
A ratio ergodic theorem for Gromov-hyperbolic groups • A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic groups preserving a σ -finite measure ( X , µ ) , was established recently by Pollicott and Sharp [2011], as follows. Wrawick, ETDS 30
A ratio ergodic theorem for Gromov-hyperbolic groups • A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic groups preserving a σ -finite measure ( X , µ ) , was established recently by Pollicott and Sharp [2011], as follows. • Let S be a symmetric generating set of such a group Γ , and let � � S n = γ ∈ Γ ; | γ | S = n , where | γ | S is the word length of γ w.r.t. S . Wrawick, ETDS 30
A ratio ergodic theorem for Gromov-hyperbolic groups • A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic groups preserving a σ -finite measure ( X , µ ) , was established recently by Pollicott and Sharp [2011], as follows. • Let S be a symmetric generating set of such a group Γ , and let � � S n = γ ∈ Γ ; | γ | S = n , where | γ | S is the word length of γ w.r.t. S . • Then there exists ρ = ρ S > 1 such that for any f , g ∈ L 1 ( X , µ ) with g > 0 : � N n = 0 ρ − n � | γ | S = n f ( γ x ) � N n = 0 ρ − n � | γ | S = n g ( γ x ) converges almost surely as N → ∞ . Limit : ???? Wrawick, ETDS 30
A ratio ergodic theorem for Gromov-hyperbolic groups • A (weighted) ratio ergodic theorem for actions of Gromov-hyperbolic groups preserving a σ -finite measure ( X , µ ) , was established recently by Pollicott and Sharp [2011], as follows. • Let S be a symmetric generating set of such a group Γ , and let � � S n = γ ∈ Γ ; | γ | S = n , where | γ | S is the word length of γ w.r.t. S . • Then there exists ρ = ρ S > 1 such that for any f , g ∈ L 1 ( X , µ ) with g > 0 : � N n = 0 ρ − n � | γ | S = n f ( γ x ) � N n = 0 ρ − n � | γ | S = n g ( γ x ) converges almost surely as N → ∞ . Limit : ???? • When Γ is the fundamental group of a closed surface of constant negative curvature with genus > 1, and the set of generators is the � � standard one, the limit is X fd µ/ X gd µ provided that the action is ergodic. Wrawick, ETDS 30
Balls on the free group • Consider the free group F = � a 1 , . . . , a r � , and let | g | denote its word length with respect to the free generators. Wrawick, ETDS 30
Balls on the free group • Consider the free group F = � a 1 , . . . , a r � , and let | g | denote its word length with respect to the free generators. • Let ( T g ) g ∈ F be a measure preserving action on a standard σ -finite measure space ( X , B , λ ) . For u , v ∈ L 1 ( X , λ ) with v > 0 let Wrawick, ETDS 30
Balls on the free group • Consider the free group F = � a 1 , . . . , a r � , and let | g | denote its word length with respect to the free generators. • Let ( T g ) g ∈ F be a measure preserving action on a standard σ -finite measure space ( X , B , λ ) . For u , v ∈ L 1 ( X , λ ) with v > 0 let � | g |≤ n T g u R n [ u , v ] := | g |≤ n T g v . � Wrawick, ETDS 30
Balls on the free group • Consider the free group F = � a 1 , . . . , a r � , and let | g | denote its word length with respect to the free generators. • Let ( T g ) g ∈ F be a measure preserving action on a standard σ -finite measure space ( X , B , λ ) . For u , v ∈ L 1 ( X , λ ) with v > 0 let � | g |≤ n T g u R n [ u , v ] := | g |≤ n T g v . � • Does the sequence R 2 n [ u , v ] of ball ratios converge pointwise almost everywhere as n → ∞ ? Wrawick, ETDS 30
Balls on the free group • Consider the free group F = � a 1 , . . . , a r � , and let | g | denote its word length with respect to the free generators. • Let ( T g ) g ∈ F be a measure preserving action on a standard σ -finite measure space ( X , B , λ ) . For u , v ∈ L 1 ( X , λ ) with v > 0 let � | g |≤ n T g u R n [ u , v ] := | g |≤ n T g v . � • Does the sequence R 2 n [ u , v ] of ball ratios converge pointwise almost everywhere as n → ∞ ? • This was recently answered negatively by Hochman [2012]. The counterexamples constructed exploit the failure of the Besicovich property for balls, and are quite general. Wrawick, ETDS 30
Recommend
More recommend
