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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.
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
Linking options:
https://www.mathnet.ru/eng/nd733 https://www.mathnet.ru/eng/nd/v16/i4/p625
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