Heterodimensional cycles of hyperbolic ergodic measures

2024/05/21 by Christian Bonatti, Bonatti, Christian, Lorenzo J. Díaz +3
Mathematics · #28A33 #37C29 #37C40 #37D30 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2405.12686

openalex publication_date 2024/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

We introduce the concept of a heterodimensional cycle of hyperbolic ergodic measures and a special type of them that we call rich. Within a partially hyperbolic context, we prove that if two measures are related by a rich heterodimensional cycle, then the entire segment of probability measures linking them lies within the closure of measures supported on periodic orbits. Motivated by the occurrence of robust heterodimensional cycles of hyperbolic basic sets, we study robust rich heterodimensional cycles of measures providing a framework for this phenomenon for diffeomorphisms. In the setting of skew products, we construct an open set of maps having uncountably many measures related by rich heterodimensional cycles.

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