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2013
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# Clémence Labrousse

175 rue du Chevaleret, 75013 Paris, France
Institut de Mathématiques de Jussieu, UMR 7586, Analyse algébrique

## 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
 Labrousse C. Flat Metrics are Strict Local Minimizers for the Polynomial Entropy 2012, vol. 17, no. 6, pp.  479-491 Abstract As we have proved in [11], the geodesic flows associated with the flat metrics on $\mathbb{T}^2$ minimize the polynomial entropy $h_{pol}$. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated with flat metrics are local strict minima for $h_{pol}$. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems. Keywords: geodesic flows, polynomial entropy, integrable systems Citation: Labrousse C.,  Flat Metrics are Strict Local Minimizers for the Polynomial Entropy, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 479-491 DOI:10.1134/S1560354712060019