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Propriétés dynamiques des difféomorphismes de l'anneau et du tore / Patrice Le Calvez

Auteur principal : Le Calvez, Patrice, 1958-, AuteurType de document : MonographieCollection : Astérisque, 204Langue : français.Pays : France.Éditeur : Paris : Société Mathématique de France, 1991Description : 1 vol. (131 p.) : ill. ; 24 cmISBN : 03031179.ISSN : 0303-1179.Bibliographie : Bibliogr. p. 121-127. Index.Sujet MSC : 37Bxx, Dynamical systems and ergodic theory, Topological dynamics
37C70, Dynamical systems and ergodic theory -- Smooth dynamical systems: general theory, Attractors and repellers, topological structure
54H20, General topology -- Connections with other structures, applications, Topological dynamics
39B12, Difference and functional equations -- Functional equations and inequalities, Iteration theory, iterative and composite equations
En-ligne : Résumé
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Séries SMF 204 (Browse shelf) Available 10743-01

Bibliogr. p. 121-127. Index

Monotone twist maps naturally appear in the theory of conservative as well as dissipative dynamical systems. The author of the present monograph gives a survey on approaches to the theory of these maps and explains several generalizations. In the first chapter the author deals with the theory of monotone twist maps. At the beginning he discusses several examples in order to motivate and illustrate results and central ideas of the theory. After recalling basic facts and definitions he explains the Aubry-Mather theory. This is a variational approach appropriate especially for the conservative case leading to criteria for the existence of periodic orbits. Then the author focuses his interest on the Birkhoff theory which is suitable for both conservative as well as dissipative systems. He applies this topological approach in order to obtain a precise description of the dynamics on invariant curves. After representing connections to the KAM-theory he studies area decreasing maps having so-called Birkhoff attractors. The second chapter bases on the fact that each diffeomorphism of the closed annulus isotopic to the identity can be written as a composition of some monotone twist maps. Hence the author uses generalizations of some of the methods described in the first chapter for studying this kind of diffeomorphisms. Firstly, he proves a version of the Poincaré- Birkhoff theorem and one of the Conley-Zehnder theorem. In both cases the periodic orbits are of a most simple as possible braid type. Then he establishes an equivariant version of the Brouwer translation theorem enabling new proofs of some results concerning the rotation set and periodic orbits of diffeomorphisms of the circle. (Zentralblatt)

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