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

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

    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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