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The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth

  • Fei Liu
  • Fang Wang
  • Weisheng WuEmail author
Article

Abstract

In this article, we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth. We prove that on each energy level E(x, v) = k with k > c(L), where c(L) is Mañé’s critical value, the Euler-Lagrange flow has positive topological entropy. This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.

Keywords

Euler-Lagrange flow positive topological entropy fundamental group exponential growth 

MSC(2010)

37B40 37D40 37J50 

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Notes

Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 11301305 and 11571207). The second author was supported by the State Scholarship Fund from China Scholarship Council (CSC). The third author was supported by National Natural Science Foundation of China (Grant No. 11701559), and the Fundamental Research Funds for the Central Universities (Grant No. 2018QC054). The second and third authors were supported by National Natural Science Foundation of China (Grant No. 11571387).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  3. 3.Department of Applied MathematicsChina Agricultural UniversityBeijingChina

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