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

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 $G_\delta$ 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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https://hal.archives-ouvertes.fr/hal-00865723
Contributor : Maxime Zavidovique <>
Submitted on : Wednesday, September 25, 2013 - 8:47:10 AM
Last modification on : Wednesday, December 9, 2020 - 3:10:30 PM

### Identifiers

• HAL Id : hal-00865723, version 1
• ARXIV : 1309.6076

### Citation

Marc Arcostanzo, Marie-Claude Arnaud, Philippe Bolle, Maxime Zavidovique. Tonelli Hamiltonians without conjugate points and $C^0$ integrability. 2013. ⟨hal-00865723⟩

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