Consider a measure-preserving action Γ y (X,μ) of a countable group Γ and a measurable cocycle α : X × Γ → Aut(Y ) with countable image, where (X,μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ y X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence...

T h e purpose of the present paper is to study the metric entropy h((j),J\r) of an F-measure preserving transformation relative to a cr-algebra M of an F-dynamical system. Concepts of sufficient families and generators are introduced and a few results are proved. Finally, the entropy of the inverse limit of an inverse spectrum of F-dynamical systems is obtained.

We show that volume-preserving perturbations of some product actions of property (T) groups exhibit a “foliation rigidity” property, which reduces the partially hyperbolic action to a family of hyperbolic actions. This is used to show that certain partially hyperbolic actions are locally rigid.

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...

The uniform norm of the differential of the n-th iteration of a diffeomorphism is called the growth sequence of the diffeomorphism. In this paper we show that there is no lower universal growth bound for volume preserving diffeomorphisms on manifolds with an effective T2 action by constructing a set of volume-preserving diffeomorphisms with arbitrarily slow growth.

A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0, ∞) is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest.

We introduce the notion of ∧and ∨-pairs of functions on lattices as an abstraction of the notions of metric and its related entropy for probability distributions. This approach allows us to highlight the relationships that exist between various properties of metrics and entropies and opens the possibility of extending these concepts to other algebraic structures.

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 ...

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 on the probability space such that the 1’st cohomology group of σΛ, H (σΛ), is equal to Char(G)×Λ. We deduce that G has un...