We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary “critical” region containing a smooth curve of tangencies between certain canonical “stable” foliations.

Let M by a complete hyperbolic 3-manifold homotopy equivalent to a compact surface Σ. Let Φ be a proper subsurface of Σ, whose boundary is sufficiently short in M . We show that the union of all Margulis tubes and cusps homotopic into Φ lifts to a uniformly quasiconvex subset of hyperbolic 3-space. 2010 Mathematics Subject Classification: 57M50

For a uniformly hyperbolic diffeomorphism f , the induced volume deformation φu in the unstable subbundle over a compact f -invariant set significantly characterizes the geometry of the set as well as the dynamics in its neighborhood. Under the hypothesis of uniform hyperbolicity, and particularly for hyperbolic surface diffeomorphisms, a large number of dynamical quantifiers such as, for examp...

Under the condition that unstable manifolds are one dimensional, the derivative formula of the potential function of the generalized SRB measure with respect to the underlying dynamical system is extended from the hyperbolic attractor case to the general case when the hyperbolic set intersecting with unstable manifolds is a Cantor set. It leads to derivative formulas of objects and quantities t...

In this paper, we prove an inequality, which we call " Devroye inequality " , for a large class of non-uniformly hyperbolic dynamical systems (M, f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and Hénon-like maps. Devroye inequality provides an upper bound for the variance of ...

We establish bounds for the measure of deviation sets associated to continuous observables with respect to weak Gibbs measures. Under some mild assumptions, we obtain upper and lower bounds for the measure of deviation sets of some non-uniformly expanding maps, including quadratic maps and robust multidimensional non-uniformly expanding local diffeomorphisms.

For every C-small perturbation B, we prove that the map (x, y) 7→ (x +a+2y, 0)+B(x, y) preserves a physical, SRB probability, for a Lebesgue positive set of parameters a. When the perturbation B is zero, this is the Jackobson theorem; when the perturbation is a small constant times (0, 1), this is the celebrated Benedicks-Carleson theorem. In particular, a new proof of the last theorems is give...

We prove that Hilbert geometries on uniformly convex Euclidean domains with C 2-boundaries are roughly isometric to the real hyperbolic spaces of corresponding dimension.