An effective version of Katok's horseshoe theorem for conservative C² surface diffeomorphisms

2017/03/18 by Bassam Fayad, Zhiyuan Zhang, Fayad, Bassam +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.1703.06254

openalex publication_date 2017/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For area preserving C² surface diffeomorphisms, we give an explicit finite information condition, on the exponential growth of the number of Bowen's (n,δ)-balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than 3.

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