Optimal results in Lorentzian Aubry\u2013Mather theory

Stefan Suhr

doi:10.1007/s00229-020-01267-2

Abstract

This article complements the Lorentzian Aubry–Mather Theory in Suhr (Geom Dedicata 160:91–117, 2012; J Fixed Point Theory Appl 21:71, 2019) by giving optimal multiplicity results for the number of maximal invariant measures. As an application the optimal Lipschitz continuity of the time separation on the Abelian cover is established.

Highlights

The structure of recurrent sets is central to the theory of dynamical systems
For geodesic flows in the absence of curvature or symmetry assumptions or other Tonelli Lagrangian flows the two main methods of producing recurrent sets are via closed orbits and minimal invariant measures in Aubry–Mather theory
In the example of flat tori every invariant set of the geodesic flow is a union of sets of this sort

Summary

Introduction

The structure of recurrent sets is central to the theory of dynamical systems. For geodesic flows in the absence of curvature or symmetry assumptions or other Tonelli Lagrangian flows the two main methods of producing recurrent sets are via closed orbits and minimal invariant measures in Aubry–Mather theory. In the example of flat tori every invariant set of the geodesic flow is a union of sets of this sort For compact manifolds both closed orbits and minimal invariant measures are known to exist. For a large class of Lorentzian 2-tori it is known though, that they admit many variationally characterized recurrent subsets outside the lightcones, see [10,14,15,17] These subsets are the support of maximal invariant measures studied in Aubry–Mather theory. If the stable time separation vanishes somewhere outside 0 there exist infinitely many ergodic maximal probability measures supported entirely in the timelike tangent vectors by Corollary 2.16, generalizing the results of [18].

Uniformly vicious spacetimes

Class A1 spacetimes

Lipschitz continuity of the time separation

Uniformly vicious Lorentzian manifolds