Measure rigidity for actions of

Abstract. A distortion theory is developed for S−unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S−unimodal maps is classified according to a

Measure Equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable free-factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups.

subspace associated to the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition to be a measurable eigenvalue. Then we consider two families

This self-contained paper is part of a series [FF1, FF2] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante-Thurston proved that every nilpotent subgroup of Diff 2 (S 1) is abelian. One of our main results is a

Abstract. We study the existence, structure, and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern

A simpler treatment is given of the theorems of Dye and Krieger concerning the classification of non-singular ergodic trans-formations up to orbit equivalence.

Abstract. In this paper we study the dynamics on T2 and C2 of a two dimensional Blaschke product. We prove that in the case that the Blaschke product is a diffeomorphism of T2 with all periodic points hyperbolic then the dynamics is hyperbolic. If a two dimensional Blaschke product diffeomorphism

Let H be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that H has compact closure in moduli space if and only if neither attractor is a fixed point.

direction we show that an action of a Polish group is whirly (“ergodic at the identity”) if and only if it admits no spatial factors, and that all actions of a Lévy group are whirly. We also show that in the Polish group Aut (X,X, µ), for the generic automorphism T the action of the subgroup Λ(T) = cls