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arXiv · 1212.3820

Non-uniform hyperbolicity and existence of absolutely continuous invariant measures

Javier Solano
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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.

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Javier Solano. 2012-12-16. Non-uniform hyperbolicity and existence of absolutely continuous invariant measures. https://arxiv.org/abs/1212.3820

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