Tonelli Hamiltonians without conjugate points and C⁰ integrability

2013/09/24 by Marc Arcostanzo, Marie-Claude Arnaud, Arcostanzo, Marc +5
Physics and Astronomy · Mathematics · #Quantum chaos and dynamical systems #Mathematical Dynamics and Fractals #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1309.6076

Abstract

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle T^*\Tⁿ of the n-dimensional torus that have no conjugate points are C⁰ integrable, i.e. T^*\Tⁿ is C⁰ foliated by a family \Fc of invariant C⁰ Lagrangian graphs. Assuming that the Hamiltonian is C^∞, we prove that there exists a G_δ subset \Gc of \Fc such that the dynamics restricted to every element of \Gc is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every C⁰ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.

Citations

• Action minimizing invariant measures for positive definite Lagrangian systems

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