Horseshoes for diffeormorphisms preserving hyperbolic measures

TL;DR

This paper extends Katok's horseshoe construction to certain diffeomorphisms preserving hyperbolic measures, providing new proofs and insights into hyperbolic dynamics.

Contribution

It offers new extensions of horseshoe constructions for $C^{1+eta}$ and $C^1$ diffeomorphisms preserving hyperbolic measures, with a self-contained proof.

Findings

Extended Katok's horseshoe construction to broader classes of diffeomorphisms.

Provided a self-contained proof of the horseshoe construction.

Clarified the role of dominated splitting in hyperbolic measure preservation.

Abstract

We give extensions of Katok's horseshoe constructions, comment on related results, and provide a self-contained proof. We consider either a $C^{1+\alpha}$ diffeomorphism preserving a hyperbolic measure or a $C^1$ diffeomorphism preserving a hyperbolic measure whose support admits a dominated splitting.