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arXiv · 1309.6076

Tonelli Hamiltonians without conjugate points and $C^0$ integrability

Marc Arcostanzo·Marie-Claude Arnaud·Philippe Bolle·Maxime Zavidovique
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Abstract

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\T^n$ of the $n$-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. $T^*\T^n$ is $C^0$ foliated by a family $\Fc$ of invariant $C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, 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 $C^0$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.

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Marc Arcostanzo, Marie-Claude Arnaud, Philippe Bolle, Maxime Zavidovique. 2013-09-24. Tonelli Hamiltonians without conjugate points and $C^0$ integrability. https://arxiv.org/abs/1309.6076

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