On density of positive Lyapunov exponents for $C^1$ symplectic diffeomorphisms

TL;DR

This paper demonstrates that symplectic diffeomorphisms with zero Lyapunov exponents can be approximated by those with positive Lyapunov exponents on a set of positive measure, showing density of such systems.

Contribution

It proves the density of symplectic diffeomorphisms with positive Lyapunov exponents in the space of all such diffeomorphisms with zero exponents.

Findings

Density of systems with positive Lyapunov exponents

Approximation of zero exponent systems by positive exponent systems

Positive measure subset with positive Lyapunov exponent

Abstract

Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $\omega$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$ approximated by one with a positive Lyapunov exponent for a positive-measured subset of $M$. That is, the set \[ \left\{ f\in \mathcal{S}ym^1_{\omega}(M)\,| \begin{array}{ll} &\mbox{The largest Lyapunov exponent }\lambda_1(f,\,x)>0\\ &\mbox{ for a positive measure set } \end{array} \right\} \] is dense in $\mathcal{S}ym^1_{\omega}(M)$. \end{abstract} \end{center}