Lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions

Marie-Claude Arnaud (1)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon

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 & 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 m; -the size of the smallest positive Lyapunov exponent of m; -the tangent cone to the support of m.

Keywords: discrete weak KAM theory; symplectic twist maps; Lyapunov exponents; Aubry-Mather theory; minimizing measures; Green bundles

Domains: Dynamical Systems [math.DS]

Marie-Claude Arnaud. Lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions. Communications in Mathematical Physics, 2016, 343 (3), pp.783-810. ⟨hal-01065120⟩