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Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus
Published online by Cambridge University Press: 19 March 2015
Abstract
We study the Euler–Lagrange flow of a Tonelli Lagrangian on the 2-torus $\mathbb{T}^{2}$ at a fixed energy level
${\mathcal{E}}\subset T\mathbb{T}^{2}$ strictly above Mañé’s strict critical value. We prove that, if for some rational direction
${\it\zeta}\in S^{1}$ there is no invariant graph
${\mathcal{T}}\subset {\mathcal{E}}$ over
$\mathbb{T}^{2}$ for the Euler–Lagrange flow with the property that all orbits on
${\mathcal{T}}$ have an asymptotic direction equal to
${\it\zeta}$, then there are chaotic dynamics in
${\mathcal{E}}$. This implies that, if the topological entropy of the Euler–Lagrange flow in
${\mathcal{E}}$ vanishes, then in
${\mathcal{E}}$ there are invariant graphs for all asymptotic directions
${\it\zeta}\in S^{1}$ and integrable-like behavior on a large scale.
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- Research Article
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- © Cambridge University Press, 2015
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