Ergodic theory of affine isometric actions on Hilbert spaces Amine - PowerPoint PPT Presentation

Ergodic theory of affine isometric actions on Hilbert spaces Amine Marrakchi (based on a joint work with Y. Arano, Y. Isono and a joint work with S. Vaes) UMPA CNRS - ENS Lyon October 5, 2020, CIRM Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Isometric action on Hilbert spaces Let H be a real Hilbert space. Every isometry of H is affine . We denote by Isom( H ) = H ⋊ O ( H ) the isometry group of H . An isometric action α : G � H of a group G is a morphism α : G → Isom( H ). It must be of the form α g ( ξ ) = π ( g ) ξ + c ( g ) where π : G → O ( H ) is an orhtogonal representation and c : G → H is a 1 -cocycle , i.e. c ( gh ) = c ( g ) + π ( g ) c ( h ) . α : G � H has a fixed point if and only if c is a coboundary , i.e. c ( g ) = ξ − π ( g ) ξ for some ξ ∈ H . Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Why study isometric actions on Hilbert spaces? Natural extension of representation theory . Interesting isometric actions on Hilbert spaces can be constructed from geometric data such as actions on trees (or some other negatively curved spaces). A countable group G has Kazhdan’s property (T) if and only if every isometric action of G on a Hilbert space has a fixed point . Examples : SL ( n , Z ) , n ≥ 3. A countable group G has the Haagerup property if and only if G admits a proper isometric action on a Hilbert space. Examples : SU ( n , 1), F n , n ≥ 2. ⇒ applications to geometric group theory , approximation properties , fixed point theorems , group cohomology , representation theory ... Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

The Gaussian distribution The standard Gaussian probability measure on R n is given by � � 1 − 1 2 � x � 2 d µ n = √ n exp d x 2 π ξ : R n → R given by For every ξ ∈ R n , the random variable � � ξ ( x ) = � ξ, x � has a Gaussian distribution of variance � ξ � 2 � � t 2 1 √ exp − . 2 � ξ � 2 � ξ � 2 π We have � � ξ � L 2 ( R n ,µ n ) = � ξ � so that R n ∋ ξ �→ � ξ ∈ L 2 ( R n , µ n ) is a linear isometry . Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Existence and uniqueness of the Gaussian process There is no analog of the Lebesgue measure on an infinite dimensional hilbert space. But there is an analog of the Gaussian probability space : Theorem (Existence and uniqueness of the Gaussian process) Let H be a separable real Hilbert space . Then there exists a standard probability space ( � H , µ ) and a linear isometry H ∋ ξ �→ � ξ ∈ L 2 ( � H , µ ) such that : � ξ has a Gaussian distribution for all ξ ∈ H. The family of random variables ( � ξ ) ξ ∈ H generates the σ -algebra of ( � H , µ ) . Moreover, the random process ( � H , µ, ( � ξ ) ξ ∈ H ) is unique , up to a unique measure preserving isomorphism . Remark : Cov ( � ξ, � η ) = � ξ, η � . Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Gaussian actions Let ( � H , µ, ( � ξ ) ξ ∈ H ) be the Gaussian process associated to H . Proposition For every U ∈ O ( H ) , there exists a unique � U ∈ Aut( � H , µ ) such that � ξ ◦ � U − 1 = � U ξ for all ξ ∈ H. The map U �→ � U is a homomorphism from O ( H ) to Aut( � H , µ ) . Think of � ξ, U − 1 η � = � U ξ, η � in the finite dimentional case! Proof : For every U ∈ O ( H ), the triple ( � H , µ, ( � U ξ ) ξ ∈ H ) is a new realization of the Gaussian process. [Connes-Weiss 1980] If π : G → O ( H ) is a representation of a group G on H , we obtain a probability measure preserving π : G → Aut( � action � H , µ ). It is called a Gaussian action . Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Example Let G be a countable group, H = ℓ 2 ( G ) and let π : G → O ( ℓ 2 ( G )) be the regular representation . We can realize the Gaussian process by taking ( � H , µ ) = ( R , ν ) ⊗ G where ν is the standard Gaussian probability measure on R . π : G � ( � Then � H , µ ) is simply the Bernoulli shift G � ( R , ν ) ⊗ G . The most studied action in ergodic theory! By taking other representations instead of the regular one, we can produce a new and rich class of p.m.p. actions . Proposition π : G � ( � If π : G → O ( H ) is a representation then � H , µ ) is ergodic if and only if π has no finite dimensional subrepresentation. Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Action of translations Proposition (Cameron-Martin 1944, Arano-Isono-M. 2019) For every v ∈ H, there exists a unique � T v ∈ Aut( � H , [ µ ]) such that ξ ◦ � � = � T − 1 ξ − � ξ, v � for all ξ ∈ H. v We have � � − 1 2 � v � 2 + � µ v := ( � T v ) ∗ µ = exp v · µ. Think of the finite dimensional case : we have � ξ, η − v � = � ξ, η � − � ξ, v � and µ v is the Gaussian probability measure centered at v � � d µ v 2 � η � 2 = exp − 1 d µ ( η ) = e − 1 1 2 � v � 2 + � v , η � 2 � η − v � 2 e . Proof : ( � H , µ v , ( � ξ − � ξ, v � ) ξ ∈ H ) is a new realization of the Gaussian process. Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Nonsingular gaussian actions The map Isom( H ) = H ⋊ O ( H ) ∋ T v ◦ U �→ � T v ◦ � U ∈ Aut( � H , [ µ ]) is a group homomorphism . For every isometric action α : G → Isom( H ), we get a nonsingular Gaussian action α : G � ( � � H , µ ). Fact: � α admits an invariant probability measure ( ≪ µ ) if and only if α has a fixed point . In that case, � α is conjugate to a classical Gaussian action . If G does not have property (T), it admits isometric actions without fixed point . ⇒ a new and rich class of nonsingular actions for all groups without property (T) . Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Ergodic properties π : G � ( � Fact: for π : G → O ( H ), we saw that � H , µ ) is ergodic if and only if π is weakly mixing ( ⇔ π has no finite dimensional subrepresentations). This is no longer true for nonsingular Gaussian actions! For every isometric action α : G � H given by α g ( ξ ) = π ( g ) ξ + c ( g ) and every t ∈ R , define a new isometric action α t : G � H by α t g ( ξ ) = π ( g ) ξ + tc ( g ) . α t : G � ( � The ergodic properties of � H , µ ) can change drastically when t varies, with a sharp phase transition phenomenon. Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Examples Let ( X , d ) be a metric space. We say that ( X , d ) is of negative type , if there exists an embedding ι : X → H such that d ( x , y ) = � ι ( x ) − ι ( y ) � 2 for all x , y ∈ X . Examples: Trees. Real hyperbolic spaces H n . Remark : if d ( x , y ) + d ( y , z ) = d ( x , z ) then ι ( x ) − ι ( y ) is orthogonal to ι ( y ) − ι ( z ). Assume that the affine subspace spanned by ι ( X ) is dense in H . Under this condition the embedding ι is unique up to a unique affine isometry! Conclusion : there exists a unique isometric action α : Isom( X , d ) � H such that α g ( ι ( x )) = ι ( g · x ) for all x ∈ X and all g ∈ Isom( X , d ). Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

The case of trees Theorem (Arano-Isono-M. 2019) Let T be a tree of bounded degree and let Γ < Aut( T ) be a discrete nonelementary subgroup. Let α : Γ � H be the associated isometric action. Then ∃ t c ∈ ]0 , + ∞ [ such that: α t : Γ � ( � The actions � H , µ ) are ergodic of type III 1 and pairwise nonconjugate for all t ∈ ]0 , t c [ . α t has a fundamental domain for all t ∈ ] t c , + ∞ [ (in � particular, it has many invariant sets and invariant σ -finite measures ). � t c = 2 2 δ (Γ) where δ (Γ) is the Poincar´ e exponent of Γ : 1 δ (Γ) = lim sup R log |{ g ∈ Γ | d ( gx 0 , x 0 ) ≤ R }| , x 0 ∈ T R → + ∞ � e − sd ( gx 0 , x 0 ) < + ∞} = inf { s > 0 | g ∈ G Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Recurrence Definition A nonsingular action σ : G � ( X , µ ) is called recurrent if for every A ⊂ X with µ ( A ) > 0, the set { g ∈ G | µ ( gA ∩ A ) > 0 } is infinite . Fact: σ is recurrent if and only if � dg ∗ µ = + ∞ almost surely . d µ g ∈ G σ is dissipative (has a fundamental domain ) if and only if � dg ∗ µ < + ∞ almost surely . d µ g ∈ G In general : X decomposes into a recurrent part and a dissipative part. Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Recurrence for Gaussian actions Let α g : ξ �→ π g ( ξ ) + c ( g ) be an isometric action of G on H . α t is recurrent if and only if Then � � � � − t 2 2 � c ( g ) � 2 + t � exp c ( g ) = + ∞ almost surely . g ∈ G Theorem (Arano-Isono-M. 2019) Let α : G � H be an isometric action. Then there exists α t is recurrent for all 0 ≤ t < t rec and t rec ∈ [0 , + ∞ ] such that � has a fundamental domain for all t > t rec . We have √ √ 2 δ ≤ t rec ≤ 2 2 δ where 1 R log |{ g ∈ G | � c ( g ) � 2 ≤ R }| δ = lim sup R → + ∞ � e − s � c ( g ) � 2 < + ∞} = inf { s > 0 | g ∈ G Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces