Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting

TL;DR

This paper establishes bounds relating the Lyapunov exponents of twisting dynamics to the angle between stable and unstable bundles, providing insights into hyperbolic measures in Hamiltonian systems.

Contribution

It introduces bounds connecting Lyapunov exponents with the angle of Oseledet's splitting for locally minimizing measures in twist maps and Hamiltonian flows.

Findings

The mean angle bounds the sum of positive Lyapunov exponents.

The angle provides a lower bound for the smallest positive Lyapunov exponent.

More precise bounds are also derived.

Abstract

We consider locally minimizing measures for the conservative twist maps of the $d$-dimensional annulus or for the Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic such measures (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and the unstable Oseledet's bundles gives an upper bound of the sum of the positive Lyapunov exponents and a lower bound of the smallest positive Lyapunov exponent. Some more precise results are proved too.