Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below.
Copy To Clipboard
Successfully Copied!
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
Front Cover for Dynamical Properties of Diffeomorphisms of the Annulus and of the Torus
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
Front Cover for Dynamical Properties of Diffeomorphisms of the Annulus and of the Torus
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.

    Readership

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

  • Additional Material
     
     
  • Request Review Copy
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.

Readership

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

Please select which format for which you are requesting permissions.