$C^1$-openness of non-uniform hyperbolic diffeomorphisms with bounded $C^2$ norm

TL;DR

This paper investigates the stability and continuity of non-uniform hyperbolicity and Lyapunov exponents within a class of $C^2$ partially hyperbolic symplectic systems that are close to the identity in $C^2$ norm, extending results to volume-preserving cases.

Contribution

It establishes $C^1$-openness of non-uniform hyperbolicity and continuity of Lyapunov exponents in bounded $C^2$ norm systems, generalizing to volume-preserving dynamics.

Findings

Non-uniform hyperbolicity is stable under $C^1$ perturbations.

Open and dense set of points with continuous center Lyapunov exponents.

Results extend to volume-preserving symplectic systems.

Abstract

We study the $C^1$-topological properties of the subset of non-uniform hyperbolic diffeomorphisms in a certain class of $C^2$ partially hyperbolic symplectic systems which have bounded $C^2$ distance to the identity. In this set, we prove the stability of non-uniform hyperbolicity as a function of the diffeomorphism and the measure, and the existence of an open and dense subset of continuity points for the center Lyapunov exponents. These results are generalized to the volume-preserving context.