Lyapunov Exponents of Minimizing Measures for Globally Positive Diffeomorphisms in All Dimensions

Abstract

The globally positive diffeomorphisms of the 2n-dimensional annulus are important because they represent what happens close to a completely elliptic periodic point of a symplectic diffeomorphism where the torsion is positive definite. For these globally positive diffeomorphisms, an Aubry–Mather theory was developed by Garibaldi and Thieullen that provides the existence of some minimizing measures. Using the two Green bundles \({G_-}\) and \({G_+}\) that can be defined along the support of these minimizing measures, we will prove that there is a deep link between: the angle between \({G_-}\) and \({G_+}\) along the support of the considered measure \({\mu}\); the size of the smallest positive Lyapunov exponent of \({\mu}\); the tangent cone to the support of \({\mu}\).