On curvature and hyperbolicity of monotone Hamiltonian systems

TL;DR

This paper investigates conditions under which monotone Hamiltonian systems exhibit hyperbolic behavior, providing characterizations of Anosov flows based on invariant distributions and curvature properties.

Contribution

It offers new criteria for Anosov behavior in monotone Hamiltonian systems using invariant distributions and curvature conditions, extending understanding of their hyperbolic dynamics.

Findings

Monotone Hamiltonian flow without conjugate points has two invariant distributions.

Anosov property is equivalent to these distributions being transversal.

Negative reduced curvature along trajectories implies Anosov behavior.

Abstract

Assume that a Hamiltonian system is monotone. In this paper, we give several characterizations on when such a system is Anosov. Assuming that a monotone Hamiltonian system has no conjugate point, we show that there are two distributions which are invariant under the Hamiltonian flow. We show that a monotone Hamiltonian flow without conjugate point is Anosov if and only if these distributions are transversal. We also show that if the reduced curvature of the Hamiltonian system is non-positive, then the flow is Anosov if and only if the reduced curvature is negative somewhere along each trajectory.