Approximation property on entropies for surface diffeomorphisms

TL;DR

This paper demonstrates that for surface diffeomorphisms with positive topological entropy, one can approximate the entropy arbitrarily closely using horseshoes, extending Gan's theorem.

Contribution

It extends Gan's theorem by showing that any $C^1$ surface diffeomorphism with positive entropy can be approximated by diffeomorphisms with horseshoes whose entropy closely matches the original.

Findings

Existence of horseshoes with entropy arbitrarily close to the original diffeomorphism.

Extension of Gan's theorem to a broader class of surface diffeomorphisms.

Approximation in the $C^1$ topology with preserved entropy properties.

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 the Theorem \cite[Theorem 1.1]{Gan} of Gan.