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

Abstract

In this article, we consider the topological entropy for autonomous positive definite Lagrangian systemson 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 Ma nes 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.