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We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction. arXiv:1211.5044v1 [math.DS]

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure preserving dynamical system is explicitly controlled. In uniformly hyper-bolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows, and counterexamples illustrating the necessity of suitable conditions on the cocycle.

In this paper, we study distortion in the group A of Affine Interval Exchange Transformations (AIET). We prove that any distorted element f of A, has an iterate f k that is conjugate by an element of A to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences we prove that no Baumslag-Solitar group, BS(m, n) with |m| = |n|, acts faithfully by elements of A; every finitely generated nilpotent group of A is virtually abelian and there is no distortion element in A Q , the subgroup of A consisting of rational AIETs.