Approximation property on entropies for surface diffeomorphisms

Wu, Wanlou; Liu, Jiansong

doi:10.1090/proc/14670

Approximation property on entropies for surface diffeomorphisms
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by Wanlou Wu and Jiansong Liu PDF

Proc. Amer. Math. Soc. 148 (2020), 223-233 Request permission

Abstract:

In this paper, we prove that for any $C^1$ surface diffeomorphism $f$ with positive topological entropy, there exists a diffeomorphism $g$ arbitrarily close (in the $C^1$ topology) to $f$ exhibiting a horseshoe $\Lambda$, such that the topological entropy of $g$ restricted on $\Lambda$ can arbitrarily approximate the topological entropy of $f$. This extends a classical result of Katok for $C^{1+\alpha }(\alpha >0)$ surface diffeomorphisms.

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