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

2025/04/14 by Vı́tor Araújo, Araujo, Vitor, Vilton Pinheiro +1
Engineering · #37C70. Secondary: 37C40 #Dynamical Systems (math.DS) #FOS: Mathematics #Hydraulic Fracturing and Reservoir Analysis #Hydrocarbon exploration and reservoir analysis #Primary: 37A25 #Reservoir Engineering and Simulation Methods

paper · pdf · doi:10.48550/arxiv.2504.10264

openalex publication_date 2025/04/14 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

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.

Citations

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Related

• Variety of physical measures in partially hyperbolic systems with multi 1-D centers
• Mixing for some non-uniformly hyperbolic systems
• Non uniform hyperbolicity and elliptic dynamics
• Poisson law for some nonuniformly hyperbolic dynamical systems with polynomial rate of mixing
• Physical measures on partially hyperbolic diffeomorphisms with multi 1-D centers