Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

TL;DR

This paper investigates the stability and continuity of Lyapunov exponents in a class of partially hyperbolic volume-preserving maps with a 2-dimensional center, showing openness and density results.

Contribution

It establishes that non-uniform hyperbolicity and Lyapunov exponent continuity are generic properties in this class of symplectic and volume-preserving systems.

Findings

Non-uniform hyperbolic maps form a $C^r$ open set.

There is a $C^r$ open and dense subset of continuity points for Lyapunov exponents.

Results are extended to volume-preserving systems.

Abstract

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.