We show that there are no partially hyperbolic horseshoes with positive Lebesgue measure for diffeomorphisms whose class of differentiability is higher than 1. This generalizes a classical result by Bowen for uniformly hyperbolic horseshoes.

We consider a few applications of Mañé’s C2 Connecting Lemma. These are the C2 creation of homoclinic points associated to a basic set (i.e., isolated transitive hyperbolic set), a C2 locally generic criterion to know whether a given point belongs to the stable set of hyperbolic homoclinic classes, and that measurably hyperbolic diffeomorphisms (i.e., having the closure of supports of all invar...

This paper is part of the general project of proof mining, developed by Kohlenbach. By ”proof mining” we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs. We present logical metatheorems for classes of spaces from functional analysis and hyperbolic geometry, like Gromov hyperbolic spaces, R-trees and uniformly ...

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measur...

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. show that only countably many these can be virtually torsion-free. This gives new examples spaces are not torsion-free and cannot subgroups groups.

We examine the uniqueness of equilibrium states for natural extension a topologically exact, non-uniformly expanding, local homeomorphism with Hölder continuous potential function. do this by applying general techniques developed Climenhaga and Thompson, show there is condition on decompositions which guarantees unique state exists. then how to apply these results certain partially hyperbolic a...

We study the non-linear supersymmetric hyperbolic sigma model $H^{2|2}$ on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in distance, we prove tightness of certain spin variables horospherical coordinates, uniformly pinning and size graph. The proof relies reduction to an effective model; its is logarithmic original model.

We prove several theorems that lend support for Greene's criterion for existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modiication that can work in the case that the Aubry-Mather sets have non-zero Lyapun...