Polynomial entropies for Bott integrable Hamiltonian systems

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 * ∈ {0, 1} and h pol ∈ {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.