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Positive topological entropy of Tonelli Lagrangian flows

Gonzalo Contreras, José Antônio G. Miranda, Luiz Gustavo Perona

Abstract

We study the topological entropy of the Lagrangian flow restricted to an energy level $E_{L}^{-1}(c) \subset TM$ for $ c >e_0(L)$. We prove that if the flow of the Tonelli Lagrangian $ L: M \to \mathbb{R}$, on a closed manifold of dimension $ n+1$, has a non-hyperbolic closed orbit or an infinite number of closed orbits with energy $ c>e_0(L)$ and satisfies certain open dense conditions, then there exist a smooth potential $ u: M\to \mathbb{R} $, with $ C^2$-norm arbitrarily small, such that the flow of the perturbed Lagrangian $ L_u=L-u$ restricted to $E_{L_u}^{-1}(c)$ has positive topological entropy. The proof of this result is based on an analog version of the Franks' Lemma for Lagrangian flows and Mañé's techniques on dominated splitting. As an application, we show that if $\dim (M)=2$ and $c > e_0(L)$, then $ L$ admits a $C^2$-perturbation by a smooth potential $u$, such that, the perturbed flow $φ_t^{L_u}\big{|}_{E_{L_u}^{-1}(c)}$ has positive topological entropy.

Positive topological entropy of Tonelli Lagrangian flows

Abstract

We study the topological entropy of the Lagrangian flow restricted to an energy level for . We prove that if the flow of the Tonelli Lagrangian , on a closed manifold of dimension , has a non-hyperbolic closed orbit or an infinite number of closed orbits with energy and satisfies certain open dense conditions, then there exist a smooth potential , with -norm arbitrarily small, such that the flow of the perturbed Lagrangian restricted to has positive topological entropy. The proof of this result is based on an analog version of the Franks' Lemma for Lagrangian flows and Mañé's techniques on dominated splitting. As an application, we show that if and , then admits a -perturbation by a smooth potential , such that, the perturbed flow has positive topological entropy.
Paper Structure (12 sections, 29 theorems, 137 equations)

This paper contains 12 sections, 29 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Frank's Lemma for Tonelli Lagrangian
  3. Finsler geometry and local coordinates
  4. General perturbations
  5. Generic Condition
  6. The Franks' Lemma for Tonelli Lagrangian
  7. Proof of the Theorem \ref{['Franks lemma']}
  8. Lagrangian Flow with infinitely many periodic orbits in an energy level
  9. Proof of theorem \ref{['t-per-hip']}
  10. Lagrangian flow with a non-hyperbolic closed orbit
  11. Proof of Proposition \ref{['energy with a non-hyperbolic orbit']}
  12. Tonelli Lagrangians on surfaces:

Key Result

Proposition 1

Let $L: TM\to \mathbb{R}$ be a smooth Tonelli Lagrangian. We suppose that the Lagrangian flow has a non-trivial closed orbit with energy $c$ and that this orbit is not hyperbolic. Then there is a smooth potential $u:M \to \mathbb{R}$ of $C^r$-norm arbitrarily small (with $4 \leq r$ ), such that $h_

Theorems & Definitions (38)

  • Proposition 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • Lemma 8
  • Proposition 9
  • Theorem 10
  • ...and 28 more