Jean-Pierre Marco

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber, which measures the ``infinitesimal variation'' of the dynamics between the fiber and the neighboring ones. This gives rise to an (upper semicontinous) torsion function, defined on the base of the system, which is a new $C^0$ (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum of the torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems. We examine the relevance of these results in the context of integrable Hamiltonian systems or diffeomorphisms, with the particular cases of $C^0$-integrable twist maps on the annulus and geodesic flows. Finally, we bound from below the polynomial entropy of $\ell$-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.

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.

We introduce two numerical conjugacy invariants of dynamical systems — the polynomial entropy and the weak polynomial entropy — which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants describe the polynomial growth rate of the number of balls (for the usual "dynamical" distances) of covers of the ambient space. We give explicit examples of computation of these polynomial entropies for generic Hamiltonian systems on surfaces.