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).
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