Hyperbolicity in the Volume Preserving Scenario

TL;DR

This paper extends a classical hyperbolicity result to volume-preserving diffeomorphisms, showing that generic volume-preserving systems with hyperbolic periodic points are Axiom A.

Contribution

It proves an analogous hyperbolicity result in the volume-preserving setting, expanding the scope of Hayashi's and Mañé's theorems.

Findings

Volume-preserving diffeomorphisms with hyperbolic periodic points are Axiom A.

The result generalizes classical hyperbolicity theorems to conservative systems.

Provides a new understanding of stability in volume-preserving dynamics.

Abstract

Hayashi has extended a result of Ma\~n\'e, proving that every diffeomorphism $f$ which has a $C^1$-neighborhood $\mathcal{U}$, where all periodic points of any $g\in\mathcal{U}$ are hyperbolic, it is an Axiom A diffeomorphism. Here, we prove the analogous result in the volume preserving scenario.