Non-uniform hyperbolicity and existence of absolutely continuous invariant measures

TL;DR

This paper establishes that non-uniform hyperbolicity in certain skew-products guarantees a finite set of ergodic absolutely continuous invariant measures, describing the typical long-term behavior of points.

Contribution

It extends the understanding of invariant measures by linking non-uniform hyperbolicity to the existence of absolutely continuous measures in specific dynamical systems.

Findings

Existence of finitely many ergodic absolutely continuous invariant measures.

Extension of Keller's theorem to non-uniformly hyperbolic one-dimensional maps.

Application of hyperbolicity recurrence properties to invariant measure existence.

Abstract

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller's theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly hyperbolic one dimensional maps.