$C^r$-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms

TL;DR

This paper demonstrates that certain partially hyperbolic symplectic diffeomorphisms with a 2-dimensional center can be approximated by non-uniformly hyperbolic systems, using the Invariance Principle under specific conditions.

Contribution

It introduces a method to approximate partially hyperbolic symplectic diffeomorphisms with non-uniform hyperbolicity, expanding understanding of hyperbolic behavior in these systems.

Findings

Approximation of partially hyperbolic symplectic diffeomorphisms by non-uniformly hyperbolic ones.

Application of the Invariance Principle of Avila and Viana.

Conditions involving pinching and bunching are crucial for the approximation.

Abstract

We use the Invariance Principle of Avila and Viana to prove that every partially hyperbolic symplectic diffeomorphism with 2-dimensional center bundle, and satisfying certain pinching and bunching conditions, can be $C^r$-approximated by non-uniformly hyperbolic diffeomorphisms.