Abstract:
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.
References