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2013
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# Jean-Pierre Marco

4 place Jussieu, 75252 Paris cedex 05, France
Universite Paris 6

## Publications:

 Labrousse C., Marco J. Polynomial Entropies for Bott Integrable Hamiltonian Systems 2014, vol. 19, no. 3, pp.  374-414 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 $h_{pol}$ 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. Keywords: dynamical complexity, entropy, integrability, Bott integrable Hamiltonians Citation: Labrousse C., Marco J.,  Polynomial Entropies for Bott Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 374-414 DOI:10.1134/S1560354714030083
 Marco J. Polynomial Entropies and Integrable Hamiltonian Systems 2013, vol. 18, no. 6, pp.  623-655 Abstract 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. Keywords: dynamical complexity, entropy, integrability, Morse Hamiltonians Citation: Marco J.,  Polynomial Entropies and Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2013, vol. 18, no. 6, pp. 623-655 DOI:10.1134/S1560354713060051