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Clémence Labrousse

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


Labrousse C., Marco J.
Polynomial Entropies for Bott Integrable Hamiltonian Systems
2014, vol. 19, no. 3, pp.  374-414
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
Labrousse C.
Flat Metrics are Strict Local Minimizers for the Polynomial Entropy
2012, vol. 17, no. 6, pp.  479-491
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

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