Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics

TL;DR

This paper derives a polynomial large deviation function for stationary measures of twisted Brownian motions on nonpositively curved surfaces, focusing on measures supported on nonhyperbolic closed geodesics, with implications for dynamics and billiards.

Contribution

It provides the first large deviation analysis for measures supported on nonhyperbolic geodesics, linking curvature behavior to deviation rates.

Findings

Large deviation function is polynomial, with degree depending on curvature near the geodesic.

Supports the understanding of measure concentration around nonhyperbolic geodesics.

Connects geometric curvature properties to probabilistic deviations in dynamical systems.

Abstract

We obtain a large deviation function for the stationary measures of twisted Brownian motions associated to the Lagrangians $L_{\lambda}(p,v)=\frac{1}{2}g_{p}(v,v)- \lambda\omega_{p}(v)$, where $g$ is a $C^{\infty}$ Riemannian metric in a compact surface $(M,g)$ with nonpositive curvature, $\omega$ is a closed 1-form such that the Aubry-Mather measure of the Lagrangian $L(p,v)=\frac{1}{2}g_{p}(v,v)-\omega_{p}(v)$ has support in a unique closed geodesic $\gamma$; and the curvature is negative at every point of $M$ but at the points of $\gamma$ where it is zero. We also assume that the Aubry set is equal to the Mather set. The large deviation function is of polynomial type, the power of the polynomial function depends on the way the curvature goes to zero in a neighborhood of $\gamma$. This results has interesting counterparts in one-dimensional dynamics with indifferent fixed points and convex billiards with flat points in the boundary of the billiard. A previous estimate by N. Anantharaman of the large deviation function in terms of the Peierl's barrier of the Aubry-Mather measure is crucial for our result.