J. G. Damasceno, J. G. Miranda, L. G. Perona Araújo, “A Note on Tonelli Lagrangian Systems on <nobr>$\mathbb{T}^2$</nobr> with Positive Topological Entropy on a High Energy Level”, Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 4,Pages <nobr>625

This article is cited in 4 papers

Mathematical problems of nonlinearity

A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

J. G. Damasceno^a, J. G. Miranda^b, L. G. Perona Araújo^c

^a Universidade Federal de Ouro Preto, R.Diogo de Vasconcelos, 122, Pilar, 35400-000, Ouro Preto, MG, Brasil
^b Departamento de Física, Instituto de Ciências Universidade Federal de Minas Gerais, Av. Antonio Carlos 6627, 31270-901, Belo Horizonte, MG, Brasil
^c Universidade Federal de Vicosa — Campus Florestal, Rodovia LMG 818, km 6, 35.690-000, Florestal, MG, Brasil

Abstract: In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.

Keywords: Tonelli Lagrangian system, Aubry – Mather theory, static classes.

MSC: 37B40, 37J50, 37J99

Received: 08.07.2020
Accepted: 21.10.2020

DOI: 10.20537/nd200407