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arXiv · 2504.10264

Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem

Vitor Araujo·Vilton Pinheiro
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Abstract

We show that the ergodic, topological and geometric basins coincide for hyperbolic dominated ergodic $cu$-Gibbs states, solving the ``basin problem'' for a wide class of non-uniformly hyperbolic systems. We obtain robust examples of exponential mixing physical measures for systems with multidimensional nonuniform hyperbolic dominated splitting, without uniformly expanding or contracting subbundles. Both results are a consequence of extending the construction of Gibbs-Markov-Young structures from partial hyperbolic systems to systems with only a dominated splitting, using the existence of an ``improved hyperbolic block'', with respect to Pesin's Nonuniform Hyperbolic Theory, for hyperbolic dominated measures of smooth maps, obtained through hyperbolic times and associated ``coherent schedules'' introduced by one of the coauthors.

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Vitor Araujo, Vilton Pinheiro. 2025-04-14. Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem. https://arxiv.org/abs/2504.10264

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