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

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 Mane's critical value, the EulerLagrange flow has positive topological entropy. This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.