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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 the Société Mathématique de France
Available Formats:
Softcover ISBN: 978-0-8218-1943-2
Product Code: SMFAMS/4
List Price: $28.00 MAA Member Price:$25.20
AMS Member Price: $22.40 Click above image for expanded view 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 the Société Mathématique de France Available Formats:  Softcover ISBN: 978-0-8218-1943-2 Product Code: SMFAMS/4  List Price:$28.00 MAA Member Price: $25.20 AMS Member Price:$22.40
• Book Details

SMF/AMS Texts and Monographs
Volume: 42000; 105 pp
MSC: Primary 58;

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.

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

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Volume: 42000; 105 pp
MSC: Primary 58;

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.