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Your search keyword '"Arnaud, M. C."' showing total 7 results
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7 results on '"Arnaud, M. C."'

1. Hyperbolicity for conservative twist maps of the 2-dimensional annulus

Author

Arnaud, M. -C

Subjects
Mathematics - Dynamical Systems
Abstract

These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative twist maps of the 2-dimensional annulus and focus on what happens close to the Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity and shape of the Aubry-Mather sets, behaviour close to the boundaries of the instability zones. We will also give some open questions. This course is the second part of a minicourse that was begun by R. Potrie. Some topics of the part of R. Potrie will be useful for this part. Many thanks to E. Maderna and L. Rifford for the invitation to give the mini-course and to R. Potrie for accepting to share the course with me.

Published
2015

2. Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians

Author

Arnaud, M. -C

Subjects
Mathematics - Dynamical Systems
Abstract

We prove the following rigidity result for the Tonelli Hamiltonians. Let T * M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (F\_n) be a sequence of Tonelli Hamiltonians that C^0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (G\_n) be a sequence of Hamiltonians that that C^0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F\_n , G\_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F, G}.

Published
2015

4. The non-hyperbolicity of irrational invariant curves for twist maps and all that follows

Author

Arnaud, M. -C, Berger, P, EA2151 Laboratoire de Mathématiques d'Avignon (LMA), Avignon Université (AU), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Institut universitaire de France, WeakKAM beyond Hamilton-Jacobi, ANR-12-BLAN-WKBHJ, and ANR-07-BLAN-0361,KAMFAIBLE,Hamilton-Jacobi et théorie KAM faible : à l'interface des EDP, systèmes dynamiques lagrangiens et symboliques(2007)

Subjects
invariant curves, Green bundles, Symplectic dynamics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Mathematics, instability zone, Lyapunov exponents, Dynamical Systems (math.DS), Mathematics - Dynamical Systems, Aubry-Mather theory, twist maps, Greene criterion, MSC 2010:37E40, 37J50, 37C40, 37D25, 37J05, 37H25
Abstract

The key result of this article is key lemma: if a Jordan curve $\gamma$ is invariant by a given C 1+$\alpha$ -diffeomorphism f of a surface and if $\gamma$ carries an ergodic hyperbolic probability $\mu$, then $\mu$ is supported on a periodic orbit. From this Lemma we deduce three new results for the C 1+$\alpha$ symplectic twist maps f of the annulus: 1. if $\gamma$ is a loop at the boundary of an instability zone such that f |$\gamma$ has an irrational rotation number, then the convergence of any orbit to $\gamma$ is slower than exponential; 2. if $\mu$ is an invariant probability that is supported in an invariant curve $\gamma$ with an irrational rotation number, then $\gamma$ is C 1 $\mu$-almost everywhere; 3. we prove a part of the so-called "Greene criterion", introduced by J. M. Greene in [16] in 1978 and never proved: assume that (pn qn) is a sequence of rational numbers converging to an irrational number $\omega$; let (f k (x n)) 1$\le$k$\le$qn be a minimizing periodic orbit with rotation number pn qn and let us denote by R n its mean residue R n = |1/2 -- Tr(Df qn (x n))/4

Published
2016

7. Tonelli Hamiltonians without conjugate points and $$C^0$$ integrability.

Author

Arcostanzo, M., Arnaud, M.-C., Bolle, P., and Zavidovique, M.

Abstract

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $$T^*\mathbb {T}^n$$ of the $$n$$ -dimensional torus that have no conjugate points are $$C^0$$ integrable, i.e. $$T^*\mathbb {T}^n$$ is $$C^0$$ foliated by a family $$\mathcal {F}$$ of invariant $$C^0$$ Lagrangian graphs. Assuming that the Hamiltonian is $$C^\infty $$ , we prove that there exists a $$G_\delta $$ subset $$\mathcal {G}$$ of $$\mathcal {F}$$ such that the dynamics restricted to every element of $$\mathcal {G}$$ is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $$C^0$$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish. [ABSTRACT FROM AUTHOR]

Published
2015
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