Clémence Labrousse

In this paper, we study the entropy of a Hamiltonian flow in restriction to an energy level where it admits a first integral which is nondegenerate in the sense of Bott. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies $h_{pol}$ and $h^*_{pol}$. We show that, under natural conditions on the critical levels of the Bott first integral and on the Hamiltonian function $H, h^*_{pol} \in {0,1}$ and $h_{pol} \in {0,1,2}$. To prove this result, our main tool is a semi-global desingularization of the Hamiltonian system in the neighborhood of a polycycle.

As we have proved in [11], the geodesic flows associated with the flat metrics on $\mathbb{T}^2$ minimize the polynomial entropy $h_{pol}$. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated with flat metrics are local strict minima for $h_{pol}$. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems.