Softcover ISBN:  9780821819432 
Product Code:  SMFAMS/4 
List Price:  $28.00 
MAA Member Price:  $25.20 
AMS Member Price:  $22.40 

Book DetailsSMF/AMS Texts and MonographsVolume: 4; 2000; 105 ppMSC: 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 AubryMather theory and Birkhoff theory, followed by some criteria for existence of periodic orbits without the areapreservation property. These are applied in the areadecreasing 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 AubryMather theory to such maps and presents a version of the PoincaréBirkhoff 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 ConleyZehnder 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.ReadershipGraduate students and research mathematicians interested in dynamical systems and geometry.

Additional Material

Request Review Copy
 Book Details
 Additional Material

 Request Review Copy
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 AubryMather theory and Birkhoff theory, followed by some criteria for existence of periodic orbits without the areapreservation property. These are applied in the areadecreasing 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 AubryMather theory to such maps and presents a version of the PoincaréBirkhoff 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 ConleyZehnder 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.