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Journal of Modern Dynamics (JMD)
 

An effective version of Katok's horseshoe theorem for conservative $C^2$ surface diffeomorphisms

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

 
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Bassam Fayad - IMJ-PRG, UP7D 58-56 Avenue de France, 75205 Paris Cedex 13, France (email)
Zhiyuan Zhang - IMJ-PRG, UP7D 58-56 Avenue de France, 75205 Paris Cedex 13, France (email)

Abstract: For area preserving $C^2$ surface diffeomorphisms, we give an explicit finite information condition on the exponential growth of the number of Bowen's $(n,\delta)$-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 $3$.

Keywords:  Pesin theory, Katok‚Äôs horseshoe theorem, entropy, effective hyperbolicity.
Mathematics Subject Classification:  Primary: 37D25, 37B40; Secondary: 74Q20.

Received: January 2017;      Revised: April 2017;      Available Online: June 2017.

 References