For each group G having an infinite normal subgroup with the relative property (T) (e.g. G = H ×K, with H infinite with property (T) and K arbitrary) and each countable abelian group Λ we construct free ergodic measure-preserving actions σΛ of G on the probability space such that the 1’st cohomology group of σΛ, H (σΛ, G), is equal to Char(G) × Λ. We deduce that G has uncountably many non stabl...

For each group G having an infinite normal subgroup with the relative property (T) (for instance groups of the form G = H ×K, where H is infinite with property (T) and K is arbitrary) and each countable abelian group Λ we construct free ergodic measure-preserving actions σΛ of G on the probability space such that the 1’st cohomology group of σΛ, H (σΛ, G), is equal to Char(G)×Λ. We deduce that ...

Let F2 denote the free group on two generators a and b. For any measure-preserving system (X,X , μ, (Tg)g∈F2 ) on a finite measure space X = (X,X , μ), any f ∈ L1(X), and any n > 1, define the averaging operators An f (x) := 1 4× 3n−1 ∑ g∈F2 :|g|=n f (T g x), where |g| denotes the word length of g. We give an example of a measure-preserving system X and an f ∈ L1(X) such that the sequence An f ...

Let {Zt}t≥0 be a Lévy process with Lévy measure ν and let τ(t) := ∫ t 0 g(r̃(u))du be a random clock, where g is a non-negative function and {r̃(t)}t≥0 is an ergodic diffusion independent of Z. Time-changed Lévy models of the form Xt := Zτt are known to be good models to capture several stylized features of asset prices such as leptokurtic distributions and volatility clustering. In our previous ...

Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the of $k$-rational points in $\mathbf{G}$. We determine set functions on which are positive type conjugation invariant, under assumption that is generated by its unipotent elements. An essential step proof classification $\mathbf{G}(k)$-invariant ergodic probability measures adelic solenoid natural...

Let {Zt}t≥0 be a Lévy process with Lévy measure ν and let τ(t) := ∫ t 0 g(r̃(u))du be a random clock, where g is a non-negative function and {r̃(t)}t≥0 is an ergodic diffusion independent of Z. Time-changed Lévy models of the form Xt := Zτt are able to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In our former paper ...

Informally speaking, the Furstenberg Correspondence [5, 6] shows that the “local behavior” of a dynamical system is controlled by the behavior of sufficiently large finite systems. By the local behavior of a dynamical system (X,B, μ,G), we mean the properties which can be stated using finitely many actions of G and the integral given by μ1. By a finite system, we just mean (S,P(S), c, G) where ...

Deformation spaces Hom(π,G)/G of representations of the fundamental group π of a surface Σ in a Lie group G admit natural actions of the mapping class group ModΣ, preserving a Poisson structure. When G is compact, the actions are ergodic. In contrast if G is noncompact semisimple, the associated deformation space contains open subsets containing the Fricke-Teichmüller space upon which ModΣ acts...