Vanishing asymptotic Maslov index for conformally symplectic flows

TL;DR

This paper proves the existence of invariant measures with zero asymptotic Maslov index for conformally symplectic flows on cotangent bundles, extending Mather theory without convexity assumptions.

Contribution

It establishes the existence of zero Maslov index measures in a general conformally symplectic setting, removing the need for convexity conditions.

Findings

Invariant measures with vanishing Maslov index exist for conformally symplectic flows.

Degenerate twist conditions guarantee ergodic measures with zero index.

Points with zero Maslov index are shown to exist in this framework.

Abstract

Motivated by Mather theory of minimizing measures for symplectic twist dynamics, we study conformally symplectic flows on a cotangent bundle. These dynamics are the most general dynamics for which it makes sense to look at (asymptotic) dynamical Maslov index. Our main result is the existence of invariant measures with vanishing index without any convexity hypothesis, in the general framework of conformally symplectic flows. A degenerate twist-condition hypothesis implies the existence of ergodic invariant measures with zero dynamical Maslov index and thus the existence of points with zero dynamical Maslov index.