Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting

Marie-Claude Arnaud (1)

1 LANLG - Laboratoire d'Analyse non linéaire et Géométrie

We consider locally minimizing measures for the conservative twist maps of the $d$-dimensional annulus or for the Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic such measures (i.e. measures with no zero Lyapunov exponents), we prove that the mean distance/angle between the stable and the unstable Oseledet's bundles gives an upper bound of the sum of the positive Lyapunov exponents and a lower bound of the smallest positive Lyapunov exponent. Some more precise results are proved too.

Keywords: Lyapunov exponents; twist maps; green bundles; Tonelli Hamiltonians; Oseledet's splitting; minimizing measures

Domains: Dynamical Systems [math.DS]

Marie-Claude Arnaud. Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of the Oseledet's splitting. Ergodic Theory and Dynamical Systems, 2012, pp.1-20. ⟨10.1017/S0143385712000065⟩. ⟨hal-00572334v2⟩