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

Bassam Fayad; Zhiyuan Zhang

doi:10.3934/jmd.2017017

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2017, Volume 11: 425-445. Doi: 10.3934/jmd.2017017

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

Bassam Fayad and
Zhiyuan Zhang

IMJ-PRG, UP7D 58-56 Avenue de France, 75205 Paris Cedex 13, France

Received: January 2017

Revised: April 18, 2017

Published: October 2017

BF: Supported in part by Projet ANR-15-CE40-0001.

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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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Mathematics Subject Classification: Primary: 37D25, 37B40; Secondary: 74Q20.

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Figure 1. $\mathcal{R}_1$ is the topological rectangle $abcd$; $\mathcal{R}_2$ is the topological rectangle $a'b'c'd'$. Under a hyperbolic map $G$, $ab$ is mapped to $a'b'$ and similarly $bc, cd, da$ are mapped respectively to $b'c', c'd', d'a'$.

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