Equidistribution for groups of toral automorphisms J. Bourgain A. - PowerPoint PPT Presentation

Equidistribution for groups of toral automorphisms J. Bourgain A. Furman E. Lindenstrauss S. Mozes 1 Institute for Advanced Study 2 University of Illinois at Chicago 3 Princeton and Hebrew University in Jerusalem 4 Hebrew University in Jerusalem UIC, May 2010 1/17

Basic dynamical questions General goal T : X → X homeomorphism of a compact space X Understand the distribution of x , Tx , . . . , T N x as N → ∞ . 2/17

Basic dynamical questions General goal T : X → X homeomorphism of a compact space X Understand the distribution of x , Tx , . . . , T N x as N → ∞ . Levels of understanding ◮ Equidistribution : ∀ x ∈ X , ∃ µ x ∈ P T ( X ) � N − 1 � 1 f ( T n x ) → ( f ∈ C ( X )) f ( y ) d µ x ( y ) N X n =0 2/17

Basic dynamical questions General goal T : X → X homeomorphism of a compact space X Understand the distribution of x , Tx , . . . , T N x as N → ∞ . Levels of understanding ◮ Equidistribution : ∀ x ∈ X , ∃ µ x ∈ P T ( X ) � N − 1 � 1 f ( T n x ) → ( f ∈ C ( X )) f ( y ) d µ x ( y ) N X n =0 ◮ Invariant measures : P T ( X ) = { µ ∈ P ( X ) : T ∗ µ = µ } 2/17

Basic dynamical questions General goal T : X → X homeomorphism of a compact space X Understand the distribution of x , Tx , . . . , T N x as N → ∞ . Levels of understanding ◮ Equidistribution : ∀ x ∈ X , ∃ µ x ∈ P T ( X ) � N − 1 � 1 f ( T n x ) → ( f ∈ C ( X )) f ( y ) d µ x ( y ) N X n =0 ◮ Invariant measures : P T ( X ) = { µ ∈ P ( X ) : T ∗ µ = µ } ◮ Closed Invariant sets 2/17

Toral automorphisms A ∈ SL d ( Z ) acts on T d = R d / Z d by A : x + Z d �→ Ax + Z d 3/17

Toral automorphisms A ∈ SL d ( Z ) acts on T d = R d / Z d by A : x + Z d �→ Ax + Z d Standard Example � � 2 1 A = 1 1 3/17

Toral automorphisms A ∈ SL d ( Z ) acts on T d = R d / Z d by A : x + Z d �→ Ax + Z d Standard Example � � 2 1 A = 1 1 Observation �� � � + Z d : gcd( p 1 , . . . , p d , q ) = 1 p 1 q , . . . , p d Periodic points = q 3/17

Toral automorphisms A ∈ SL d ( Z ) acts on T d = R d / Z d by A : x + Z d �→ Ax + Z d Standard Example � � 2 1 A = 1 1 Observation �� � � + Z d : gcd( p 1 , . . . , p d , q ) = 1 p 1 q , . . . , p d Periodic points = q Single hyperbolic automorphism 1 Closed Invariant sets : of every Hausdorff dim [0 , d ] 3/17

Toral automorphisms A ∈ SL d ( Z ) acts on T d = R d / Z d by A : x + Z d �→ Ax + Z d Standard Example � � 2 1 A = 1 1 Observation �� � � + Z d : gcd( p 1 , . . . , p d , q ) = 1 p 1 q , . . . , p d Periodic points = q Single hyperbolic automorphism 1 Closed Invariant sets : of every Hausdorff dim [0 , d ] 2 Invariant measures : uncountably many distinct ergodic 3/17

Toral automorphisms A ∈ SL d ( Z ) acts on T d = R d / Z d by A : x + Z d �→ Ax + Z d Standard Example � � 2 1 A = 1 1 Observation �� � � + Z d : gcd( p 1 , . . . , p d , q ) = 1 p 1 q , . . . , p d Periodic points = q Single hyperbolic automorphism 1 Closed Invariant sets : of every Hausdorff dim [0 , d ] 2 Invariant measures : uncountably many distinct ergodic 3 Equidistribution : no chance ! 3/17

Abelian groups of toral automorphisms Setup ”Non degenerate” Z k < SL d ( Z ) with 2 ≤ k ≤ d − 1 4/17

Abelian groups of toral automorphisms Setup ”Non degenerate” Z k < SL d ( Z ) with 2 ≤ k ≤ d − 1 Rigidity phenomena 4/17

Abelian groups of toral automorphisms Setup ”Non degenerate” Z k < SL d ( Z ) with 2 ≤ k ≤ d − 1 Rigidity phenomena 1 Closed Invariant sets : Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 4/17

Abelian groups of toral automorphisms Setup ”Non degenerate” Z k < SL d ( Z ) with 2 ≤ k ≤ d − 1 Rigidity phenomena 1 Closed Invariant sets : Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 2 Invariant measures : ◮ Conjecture: Atomic (rational pts) + Lebesgue 4/17

Abelian groups of toral automorphisms Setup ”Non degenerate” Z k < SL d ( Z ) with 2 ≤ k ≤ d − 1 Rigidity phenomena 1 Closed Invariant sets : Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 2 Invariant measures : ◮ Conjecture: Atomic (rational pts) + Lebesgue ◮ Positive entropy (equivalently dim H ( µ ) > 0) understood by: D. Rudolph, A. Katok, R. Spatzier, B. Host, B. Kalinin, E. Lindenstrauss, M. Einsiedler, ... 4/17

Abelian groups of toral automorphisms Setup ”Non degenerate” Z k < SL d ( Z ) with 2 ≤ k ≤ d − 1 Rigidity phenomena 1 Closed Invariant sets : Finite (rational pts), T d H. Furstenberg (77), D. Berend (84) 2 Invariant measures : ◮ Conjecture: Atomic (rational pts) + Lebesgue ◮ Positive entropy (equivalently dim H ( µ ) > 0) understood by: D. Rudolph, A. Katok, R. Spatzier, B. Host, B. Kalinin, E. Lindenstrauss, M. Einsiedler, ... 3 No equidistribution 4/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) 5/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) What is equidistribution for Γ . x ? 5/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) What is equidistribution for Γ . x ? Fix a prob meas ν on Γ with Γ = � supp ( ν ) � . Consider � µ n , x = ν ∗ n ∗ δ x = ν ( g n ) · · · ν ( g 1 ) · δ g n ··· g 1 x . 5/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) What is equidistribution for Γ . x ? Fix a prob meas ν on Γ with Γ = � supp ( ν ) � . Consider � µ n , x = ν ∗ n ∗ δ x = ν ( g n ) · · · ν ( g 1 ) · δ g n ··· g 1 x . Remark � N − 1 1 Weak-* limits of n =0 µ n , x are ν -stationary measures N � � � P ν ( X ) = µ ∈ P ( X ) : µ = ν ∗ µ = ν ( g ) · g ∗ µ 5/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) What is equidistribution for Γ . x ? Fix a prob meas ν on Γ with Γ = � supp ( ν ) � . Consider � µ n , x = ν ∗ n ∗ δ x = ν ( g n ) · · · ν ( g 1 ) · δ g n ··· g 1 x . Remark � N − 1 1 Weak-* limits of n =0 µ n , x are ν -stationary measures N � � � P ν ( X ) = µ ∈ P ( X ) : µ = ν ∗ µ = ν ( g ) · g ∗ µ ◮ P Γ ( X ) ⊆ P ν ( X ) convex compact subsets of P ( X ) 5/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) What is equidistribution for Γ . x ? Fix a prob meas ν on Γ with Γ = � supp ( ν ) � . Consider � µ n , x = ν ∗ n ∗ δ x = ν ( g n ) · · · ν ( g 1 ) · δ g n ··· g 1 x . Remark � N − 1 1 Weak-* limits of n =0 µ n , x are ν -stationary measures N � � � P ν ( X ) = µ ∈ P ( X ) : µ = ν ∗ µ = ν ( g ) · g ∗ µ ◮ P Γ ( X ) ⊆ P ν ( X ) convex compact subsets of P ( X ) ◮ P Γ ( X ) = ∅ is possible for non-amenable Γ. 5/17

Large groups of toral automorphisms Setup Γ < SL d ( Z ) which is Zariski dense in SL d ( R ) What is equidistribution for Γ . x ? Fix a prob meas ν on Γ with Γ = � supp ( ν ) � . Consider � µ n , x = ν ∗ n ∗ δ x = ν ( g n ) · · · ν ( g 1 ) · δ g n ··· g 1 x . Remark � N − 1 1 Weak-* limits of n =0 µ n , x are ν -stationary measures N � � � P ν ( X ) = µ ∈ P ( X ) : µ = ν ∗ µ = ν ( g ) · g ∗ µ ◮ P Γ ( X ) ⊆ P ν ( X ) convex compact subsets of P ( X ) ◮ P Γ ( X ) = ∅ is possible for non-amenable Γ. ◮ P ν ( X ) � = ∅ , any closed invariant set supports ν -stationary measures 5/17

Overview of the results Setup Γ < SL d ( Z ) which is Z-dense, or more generally 6/17

Overview of the results Setup Γ < SL d ( Z ) which is Z-dense, or more generally Γ strongly irreducible and Γ ∋ proximal element 6/17

Overview of the results Setup Γ < SL d ( Z ) which is Z-dense, or more generally Γ strongly irreducible and Γ ∋ proximal element Rigidity phenomena 1 Closed Γ- invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc’h-A. Starkov (04) 6/17

Overview of the results Setup Γ < SL d ( Z ) which is Z-dense, or more generally Γ strongly irreducible and Γ ∋ proximal element Rigidity phenomena 1 Closed Γ- invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc’h-A. Starkov (04) 2 Γ- invariant measures = Atomic (rational pts) + Lebesgue BFLM (07, 10), Y. Benoist-J.F. Quint (10) 6/17

Overview of the results Setup Γ < SL d ( Z ) which is Z-dense, or more generally Γ strongly irreducible and Γ ∋ proximal element Rigidity phenomena 1 Closed Γ- invariant sets = Finite (rational pts), T d R. Muchnik (05), Y. Guivarc’h-A. Starkov (04) 2 Γ- invariant measures = Atomic (rational pts) + Lebesgue BFLM (07, 10), Y. Benoist-J.F. Quint (10) 3 ν - stationary measures = Γ-invariant = Atomic + Lebesgue BLFM (07, 10), Y. Benoist-J.F. Quint (10) 6/17