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Dynamical Properties of Diffeomorphisms of the Annulus and of the Torus
Patrice Le Calvez, University of Paris, Villetaneuse, France
A co-publication of the AMS and Société Mathématique de France.
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SMF/AMS Texts and Monographs
2000; 105 pp; softcover
Volume: 4
ISBN-10: 0-8218-1943-7
ISBN-13: 978-0-8218-1943-2
List Price: US$24
Member Price: US$19.20
Order Code: SMFAMS/4
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The first chapter of this monograph presents a survey of the theory of monotone twist maps of the annulus. First, the author covers the conservative case by presenting a short survey of Aubry-Mather theory and Birkhoff theory, followed by some criteria for existence of periodic orbits without the area-preservation property. These are applied in the area-decreasing case, and the properties of Birkhoff attractors are discussed. A diffeomorphism of the closed annulus which is isotopic to the identity can be written as the composition of monotone twist maps.

The second chapter generalizes some aspects of Aubry-Mather theory to such maps and presents a version of the Poincaré-Birkhoff theorem in which the periodic orbits have the same braid type as in the linear case. A diffeomorphism of the torus isotopic to the identity is also a composition of twist maps, and it is possible to obtain a proof of the Conley-Zehnder theorem with the same kind of conclusions about the braid type, in the case of periodic orbits. This result leads to an equivariant version of the Brouwer translation theorem which permits new proofs of some results about the rotation set of diffeomorphisms of the torus.

This is the English translation of a volume previously published as volume 204 in the Astérisque series.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.

Readership

Graduate students and research mathematicians interested in dynamical systems and geometry.

Table of Contents

  • Presentation and comparison of the different approaches to the theory of monotone twist diffeomorphisms of the annulus
  • Generating phases of the diffeomorphisms of the torus and the annulus
  • Index
  • Bibliography
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