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

  • Fei Liu
  • Fang Wang
  • Weisheng WuEmail author


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.


Euler-Lagrange flow positive topological entropy fundamental group exponential growth 


37B40 37D40 37J50 


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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).


  1. 1.
    Carneiro M J D. On minimizing measures of the action of autonomous Lagrangians. Nonlinearity, 1995, 8: 1077–1085MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dinaburg E I. On the relations among various entropy characteristics of dynamical systems. Math USSR Izv, 1971, 5: 337–378CrossRefzbMATHGoogle Scholar
  3. 3.
    Egloff D. On the dynamics of uniform Finsler manifolds of negative flag curvature. Ann Global Anal Geom, 1997, 15: 101–116MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fathi A. The Weak KAM Theorem in Lagrangian Dynamics. Cambridge Studies in Advanced Mathematics, vol. 88. Cambridge: Cambridge University Press, 2008Google Scholar
  5. 5.
    Glasmachers E, Knieper G. Characterization of geodesic flows on T 2 with and without positive topological entropy. Geom Funct Anal, 2010, 20: 1259–1277MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Katok A, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge: Cambridge University Press, 1995CrossRefzbMATHGoogle Scholar
  7. 7.
    Mañé R. Lagrangian flows: The dynamics of globally minimizing orbits. Bol Soc Brasil Mat (NS), 1997, 28: 141–153MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Manning A. Topological entropy for geodesic flows. Ann of Math (2), 1979, 110: 567–573MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mather J. Action minimizing invariant probability measures for positive definite Lagrangian systems. Math Z, 1991, 207: 169–207MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mather J. Variational construction of connecting orbits. Ann Inst Fourier (Grenoble), 1993, 43: 1349–1386MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Paternain G P. Geodesic Flows. Progress in Mathematics, vol. 180. Boston: Birkhäuser, 1999CrossRefzbMATHGoogle Scholar
  12. 12.
    Shen Y B, Zhao W. On fundamental groups of Finsler manifolds. Sci China Math, 2011, 54: 1951–1964MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Walters P. An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. New York-Berlin: Springer-Verlag, 1982CrossRefzbMATHGoogle Scholar

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© 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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