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Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 4, Pages 625–635 (Mi nd733)  

This article is cited in 4 scientific papers (total 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. Damascenoa, J. G. Mirandab, L. G. Perona Araújoc

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

DOI: https://doi.org/10.20537/nd200407

Full text: PDF file (343 kB)
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MSC: 37B40, 37J50, 37J99
Received: 08.07.2020
Accepted:21.10.2020

Citation: J. G. Damasceno, J. G. Miranda, L. G. Perona Araújo, “A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level”, Rus. J. Nonlin. Dyn., 16:4 (2020), 625–635

Citation in format AMSBIB
\Bibitem{DamMirPer20}
\by J.~G.~Damasceno, J.~G.~Miranda, L.~G.~Perona Ara\'ujo
\paper A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level
\jour Rus. J. Nonlin. Dyn.
\yr 2020
\vol 16
\issue 4
\pages 625--635
\mathnet{http://mi.mathnet.ru/nd733}
\crossref{https://doi.org/10.20537/nd200407}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4198784}


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