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 αg(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

• nly 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), Fn, 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 Rn is given by dµn = 1 √ 2π

n exp

• −1

2x2

• dx

For every ξ ∈ Rn, the random variable ξ : Rn → R given by

• ξ(x) = ξ, x has a Gaussian distribution of variance ξ2

1 ξ √ 2π exp

• −

t2 2ξ2

• .

We have

• ξL2(Rn,µn) = ξ

so that Rn ∋ ξ → ξ ∈ L2(Rn, µn) 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 ∈ L2( 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 = Uξ for all ξ ∈ H. The map U → U is a homomorphism from O(H) to Aut( H, µ). Think of ξ, U−1η = Uξ, η in the finite dimentional case! Proof : For every U ∈ O(H), the triple ( H, µ, ( Uξ)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 action π : G → Aut( 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 = ℓ2(G) and let π : G → O(ℓ2(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. Then π : G ( 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 If π : G → O(H) is a representation then π : G ( 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 Tv ∈ Aut( H, [µ]) such that

• ξ ◦

T −1

v

= ξ − ξ, v for all ξ ∈ H. We have µv := ( Tv)∗µ = exp

• −1

2v2 + v

• · µ.

Think of the finite dimensional case : we have ξ, η − v = ξ, η − ξ, v and µv is the Gaussian probability measure centered at v dµv dµ (η) = e− 1

2 η−v2e 1 2 η2 = exp

• −1

2v2 + v, η

• .

Proof : ( H, µv, ( 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) ∋ Tv ◦ U → Tv ◦ 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

Fact: for π : G → O(H), we saw that π : G ( 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 αg(ξ) = π(g)ξ + c(g) and every t ∈ R, define a new isometric action αt : G H by αt

g(ξ) = π(g)ξ + tc(g).

The ergodic properties of αt : G ( 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 Hn. Remark : if d(x, y) + d(y, z) = d(x, z) then ι(x) − ι(y) is

• rthogonal 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 αg(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 ∃tc ∈]0, +∞[ such that: The actions αt : Γ ( H, µ) are ergodic of type III1 and pairwise nonconjugate for all t ∈]0, tc[.

• αt has a fundamental domain for all t ∈]tc, +∞[ (in

particular, it has many invariant sets and invariant σ-finite measures). tc = 2

• 2δ(Γ) where δ(Γ) is the Poincar´

e exponent of Γ: δ(Γ) = lim sup

R→+∞

1 R log |{g ∈ Γ | d(gx0, x0) ≤ R}|, x0 ∈ T = inf{s > 0 |

• g∈G

e−sd(gx0,x0) < +∞}

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

• g∈G

dg∗µ dµ = +∞ almost surely. σ is dissipative (has a fundamental domain) if and only if

• g∈G

dg∗µ dµ < +∞ almost surely. 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 αg : ξ → πg(ξ) + c(g) be an isometric action of G on H. Then αt is recurrent if and only if

• g∈G

exp

• −t2

2 c(g)2 + t c(g)

• = +∞ almost surely.

Theorem (Arano-Isono-M. 2019) Let α : G H be an isometric action. Then there exists trec ∈ [0, +∞] such that αt is recurrent for all 0 ≤ t < trec and has a fundamental domain for all t > trec. We have √ 2δ ≤ trec ≤ 2 √ 2δ where δ = lim sup

R→+∞

1 R log |{g ∈ G | c(g)2 ≤ R}| = inf{s > 0 |

• g∈G

e−sc(g)2 < +∞}

Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Ergodicity and type

For a group Γ < Aut(T), the associated action α : Γ H is proper and its linear part is mixing. Theorem (Arano-Isono-M. 2019) Let α : G H be an isometric action. Suppose that its linear part π = α0 is mixing, i.e. limg→∞π(g)ξ, η = 0 for all ξ, η ∈ H. Then αt is ergodic for all t < trec. Theorem (M.-Vaes 2020) Let α : G H be an isometric action that has no fixed point. Then there exists tc ∈ [0, +∞] such that:

• αt : G (

H, µ) is of type III for all 0 < t < tc. For t > tc, αt admits an invariant σ-finite measure ∼ µ. If α is proper, we have tc = trec. .

Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

A key tool : the rotation trick

Let α : G H be an isometric action. Take t, s ∈ R with t2 + s2 = 1. Define the rotation R ∈ O(H ⊕ H) by R = t −s s t

• Then for all g ∈ G, we have

(αt

g ⊕ αs g) ◦ R = R ◦ (αg ⊕ α0 g).

We can identify the Gaussian probability space of H ⊕ H with ( H × H, µ ⊗ µ). Then for all g ∈ G, we have ( αt

g ×

αs

g) ◦

R = R ◦ ( αg × α0

g).

We can relate the Gaussian actions αt for different values of t, even t = 0!

Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Beyond the mixing/proper case

[M.-Vaes 2020] If π(gn) → 0 fast enough and c(gn) does not grow too fast, then α is ergodic of type III1. We also have : Theorem (M.-Vaes 2020) Let α : Γ H be an isometric action and suppose that the linear part π = α0 is nonamenable. Then αt is strongly ergodic of type III1 for t > 0 small enough. Theorem (M.-Vaes 2020) Let Γ < H be an additive subgroup and consider the translation action α : Γ H. Suppose that Γ is sequencially weakly dense. Then αt is ergodic of type III1 for all t > 0.

Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces

Thank you for your attention!

Amine Marrakchi Ergodic theory of affine isometric actions on Hilbert spaces