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 ${\rm{h_{pol}}}$ and ${\rm{h_{pol}^*}}$. We show that, under natural conditions on the critical levels of the Bott first integral and on the Hamiltonian function $H$, ${\rm{h_{pol}^*}}\in \{0,1\}$ and ${\rm{h_{pol}}}\in \{0,1,2\}$. To prove this result, our main tool is a semi-global desingularization of the Hamiltonian system in the neighborhood of a polycycle.