Commutator criteria for strong mixing Rafael Tiedra de Aldecoa - PowerPoint PPT Presentation

Commutator criteria for strong mixing Rafael Tiedra de Aldecoa Pontifical Catholic University of Chile Prague, June 2016 Work in part with S. Richard (Nagoya University) 1 / 25

Table of Contents Strong mixing 1 Commutators 2 Discrete groups 3 Continuous groups 4 Time changes of horocycle flows 5 References 6 2 / 25

Strong mixing Strong mixing Example (Discrete group of unitary operators) If U is a unitary operator in a Hilbert space H , U n := U n , n ∈ Z , defines a discrete 1 -parameter group of unitary operators. Example (Continuous group of unitary operators) If H is a self-adjoint operator in a Hilbert space H , then U t := e − itH , t ∈ R , defines a strongly continuous 1 -parameter group of unitary operators. 3 / 25

Strong mixing Example (Koopman operator) If T : X → X is an automorphism of a probability space ( X , µ ) , then the Koopman operator U T : L 2 ( X , µ ) → L 2 ( X , µ ) , ϕ �→ ϕ ◦ T , is a unitary operator. 4 / 25

Strong mixing Ergodicity, weak mixing and strong mixing of an automorphism T : X → X are expressible in terms of the Koopman operator U T : • T is ergodic iff 1 is a simple eigenvalue of U T . • T is weakly mixing iff U T has purely continuous spectrum in { C · 1 } ⊥ . • T is strongly mixing iff � � ϕ, ( U T ) N ϕ for all ϕ ∈ { C · 1 } ⊥ . lim = 0 N →∞ strong weak a.c. spectrum in { C · 1 } ⊥ ⇒ ⇒ ergodicity ⇒ mixing mixing 5 / 25

Commutators Commutators • H , arbitrary Hilbert space with norm � · � and scalar product � · , · � • B ( H ), bounded linear operators on H • A , self-adjoint operator in H with domain D ( A ) 6 / 25

Commutators Definition An operator S ∈ B ( H ) satisfies S ∈ C k ( A ) if R ∋ t �→ e − itA S e itA ∈ B ( H ) is strongly of class C k . S ∈ C 1 ( A ) if and only if � � � � A ϕ, S ϕ � − � ϕ, SA ϕ � � ≤ Const. � ϕ � 2 for all ϕ ∈ D ( A ) . The operator corresponding to the continuous extension of the quadratic form is denoted by [ S , A ], and one has � [ iS , A ] = s- d � t =0 e − itA S e itA ∈ B ( H ) . � d t 7 / 25

Commutators Definition A self-adjoint operator H in H is of class C k ( A ) if ( H − z ) − 1 ∈ C k ( A ) for some z ∈ C \ σ ( H ). If H is of class C 1 ( A ), then � A , ( H − z ) − 1 � = ( H − z ) − 1 [ H , A ]( H − z ) − 1 , � D ( H ) , D ( H ) ∗ � with [ H , A ] ∈ B the operator corresponding to the continuous extension to D ( H ) of the quadratic form D ( H ) ∩ D ( A ) ∋ ϕ �→ � H ϕ, A ϕ � − � A ϕ, H ϕ � ∈ C . 8 / 25

Discrete groups Discrete groups Theorem (Strong mixing for discrete groups) Let U be a unitary operator in H and let A be a self-adjoint operator in H with U ∈ C 1 ( A ) . Assume that the strong limit N − 1 � � A , U N � U n � [ A , U ] U − 1 � 1 1 U − N = s-lim U − n D := s-lim N N N →∞ N →∞ n =0 exists. Then, � � = 0 for each ϕ ∈ ker( D ) ⊥ and ψ ∈ H , ϕ, U N ψ (a) lim N →∞ (b) U | ker( D ) ⊥ has purely continuous spectrum. 9 / 25

Discrete groups • D is bounded and self-adjoint because it is the strong limit of bounded self-adjoint operators. • DU n = U n D for each n ∈ Z . So, ker( D ) ⊥ is a reducing subspace for U , and U | ker( D ) ⊥ is a unitary operator. • Point (b) is a simple consequence of point (a). 10 / 25

Discrete groups Sketch of the proof of (a). ϕ ∈ D D ( A ), ψ ∈ D ( A ), N ∈ N ∗ , and Let ϕ = D � � A , U N � D N := 1 U − N . N Since U N , U − N ∈ C 1 ( A ), we have U N ψ, U − N � ϕ ∈ D ( A ). Thus, � �� �� ϕ, U N ψ � �� � � � �� � ϕ, U N ψ ϕ, U N ψ = ( D − D N ) � + D N � � � � �� ��� A , U N � � � ψ � + 1 U − N � � ( D − D N ) � ϕ, U N ψ � ≤ ϕ N � � � �� � �� �� �� � � ψ � + 1 � + 1 U N AU − N � � ( D − D N ) � ϕ, U N ψ ϕ, U N ψ � ≤ A � ϕ N N � � � � � � � � ψ � + 1 � � ψ � + 1 � ( D − D N ) � � A � �� � � A ψ � . ≤ ϕ ϕ ϕ N N � � ϕ, U N ψ Since D = s-lim N D N , we get lim N = 0, and the claim follows from density arguments. 11 / 25

Discrete groups Remark If � N ≥ 1 � ( D − D N ) ϕ � 2 < ∞ for suitable ϕ ∈ H , then � � �� �� � 2 < ∞ ϕ, U N ϕ for all ϕ ∈ ker( D ) ⊥ , N ≥ 1 and U | ker( D ) ⊥ has purely a.c. spectrum. 12 / 25

Discrete groups Example (Cocycles with values in compact Lie groups) · · · 13 / 25

Continuous groups Continuous groups Theorem (Strong mixing for continuous groups) Let H and A be self-adjoint operators in H with ( H − i ) − 1 ∈ C 1 ( A ) . Assume that � t 1 d s e isH ( H + i ) − 1 [ iH , A ]( H − i ) − 1 e − isH D := s-lim t t →∞ 0 exists. Then, � � ϕ, e − itH ψ = 0 for each ϕ ∈ H and ψ ∈ ker( D ) ⊥ , (a) lim t →∞ (b) H | ker( D ) ⊥ has purely continuous spectrum. 14 / 25

Continuous groups • The proof is similar to the one for unitary operators (just more domain issues because both H and A are unbounded). • The results in the unitary case (the group ( Z , +)) and in the self-ajoint case (the group ( R , +)) are particular cases of a more general criterion for the strong mixing property of unitary representations of topological groups. 15 / 25

Continuous groups Example (Canonical commutation relation) Assume that ( H − i ) − 1 ∈ C 1 ( A ) with [ iH , A ] = 1 . Then, for all t > 0 � t D t := 1 d s e isH ( H + i ) − 1 [ iH , A ]( H − i ) − 1 e − isH = ( H 2 + 1) − 1 = D t 0 and ker( D ) = { 0 } . So, the theorem implies that H has purely a.c. spectrum. In fact, we have in this case Weyl commutation relation e − itA e isH e itA = e ist e isH , s , t ∈ R . Thus, Stone-von Neumann theorem implies that H has Lebesgue spectrum with uniform multiplicity. 16 / 25

Time changes of horocycle flows Example (Time changes of horocycle flows). • Σ, compact Riemannian surface of constant negative curvature • M := T 1 Σ, unit tangent bundle of Σ ( M is a compact 3-manifold with probability measure µ , M ≃ Γ \ PSL(2; R ) for some cocompact lattice Γ in PSL(2; R )) • F h := { F h , t } t ∈ R , horocycle flow on M • F g := { F g , t } t ∈ R , geodesic flow on M F h and F g are one-parameter groups of diffeomorphisms preserving the measure µ . 17 / 25

Time changes of horocycle flows Geodesic flow in the Poincar´ e half plane 18 / 25

Time changes of horocycle flows Positive horocycle flow in the Poincar´ e half plane (from Bekka/Mayer’s book) 19 / 25

Time changes of horocycle flows Geodesics and horocycles in the Poincar´ e half plane (from Hasselblatt/Katok’s book) 20 / 25

Time changes of horocycle flows Each flow has an essentially self-adjoint generator ϕ ∈ C ∞ ( M ) ⊂ L 2 ( M , µ ) , H j ϕ := − iX j ϕ, with X j the vector field of F h or F g . H h is of class C 1 ( H g ) with � � iH h , H g = H h . A C 1 -time change of X h is a vector field f X h with f ∈ C 1 � � M ; (0 , ∞ ) . f X h has a complete flow � F h := { � F h , t } t ∈ R with generator H := f H h essentially self-adjoint on C 1 ( M ) ⊂ H := L 2 ( M , f − 1 µ ). 21 / 25

Time changes of horocycle flows A := f 1 / 2 H g f − 1 / 2 is self-adjoint in H , and ( H − i ) − 1 ∈ C 1 ( A ) with ( H + i ) − 1 [ iH , A ]( H − i ) − 1 = ( H + i ) − 1 � � ( H − i ) − 1 H ξ + ξ H and ξ := 1 2 − 1 2 f − 1 X g ( f ) . So, � t D t = 1 d s e isH ( H + i ) − 1 [ iH , A ]( H − i ) − 1 e − isH t 0 = ( H + i ) − 1 � � ( H − i ) − 1 H ξ t + ξ t H with � t � t � � ξ t := 1 d s e isH ξ e − isH = 1 ξ ◦ � d s F h , − s . t t 0 0 22 / 25

Time changes of horocycle flows Since F h is uniquely ergodic with respect to µ , � F h is uniquely ergodic with f − 1 µ respect to � µ := M f − 1 d µ . Thus, � � � � f − 1 � t →∞ ξ t = 1 2 − 1 µ f − 1 X g ( f ) = 1 1 = 1 � lim d � 2 + d µ X g 2 M f − 1 d µ 2 2 M M uniformly on M , and t →∞ D t = ( H + i ) − 1 � � � � − 1 . ( H − i ) − 1 = H H 2 + 1 H · 1 2 + 1 D := s-lim 2 · H So, ker( D ) = ker( H ), and the theorem implies that � � ϕ, e − itH ψ for all ϕ ∈ H and ψ ∈ ker( H ) ⊥ . lim = 0 t →∞ Therefore, C 1 -time changes of horocycle flows are strongly mixing. 23 / 25

Time changes of horocycle flows Thank you ! 24 / 25

References References • B. Marcus. Ergodic properties of horocycle flows for surfaces of negative curvature. Ann. of Math., 1977 • S. Richard and R. Tiedra de Aldecoa. Commutator criteria for strong mixing II. More general and simpler. preprint on arXiv • R. Tiedra de Aldecoa. Commutator criteria for strong mixing. Ergodic Theory Dynam. Systems, 2016 25 / 25