Positive Lyapunov exponents for Hamiltonian linear differential systems

TL;DR

This paper demonstrates that a generic set of Hamiltonian linear differential systems exhibit positive Lyapunov exponents, indicating chaos, especially over hyperbolic flows, thus answering longstanding questions in the field.

Contribution

It proves that positive Lyapunov exponents are prevalent in Hamiltonian systems over certain flows, extending understanding of chaos in these systems.

Findings

Open and dense set of Hamiltonian systems with positive Lyapunov exponents

Positive exponents occur over suspension flows with bounded roof functions

Results apply to cocycles over hyperbolic flows with ergodic measures

Abstract

In the present paper we give a positive answer to some questions posed by Viana on the existence of positive Lyapunov exponents for Hamiltonian linear differential systems. We prove that there exists an open and dense set of Hamiltonian linear differential systems, over a suspension flow with bounded roof function, displaying at least one positive Lyapunov exponent. In consequence, typical cocycles over a uniformly hyperbolic flow are chaotic. Finally, we obtain similar results for cocycles over flows preserving an ergodic, hyperbolic measure with local product structure.