# Dynamical Properties of Diffeomorphisms of the Annulus and of the Torus

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*Patrice Le Calvez*

A co-publication of the AMS and Société Mathématique de France

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.