Dynamical properties of diffeomorphisms of the annulus and the torus.
(Propriétés dynamiques des difféomorphismes de l’anneau et du tore.)

*(French)*Zbl 0784.58033
Astérisque. 204. Paris: Société Mathématique de France, 131 p. (1991).

Monotone twist maps naturally appear in the theory of conservative as well as dissipative dynamical systems. The author of the present monograph gives a survey on approaches to the theory of these maps and explains several generalizations.

In the first chapter the author deals with the theory of monotone twist maps. At the beginning he discusses several examples in order to motivate and illustrate results and central ideas of the theory. After recalling basic facts and definitions he explains the Aubry-Mather theory. This is a variational approach appropriate especially for the conservative case leading to criteria for the existence of periodic orbits. Then the author focuses his interest on the Birkhoff theory which is suitable for both conservative as well as dissipative systems. He applies this topological approach in order to obtain a precise description of the dynamics on invariant curves. After representing connections to the KAM-theory he studies area decreasing maps having so-called Birkhoff attractors.

The second chapter bases on the fact that each diffeomorphism of the closed annulus isotopic to the identity can be written as a composition of some monotone twist maps. Hence the author uses generalizations of some of the methods described in the first chapter for studying this kind of diffeomorphisms. Firstly, he proves a version of the Poincaré- Birkhoff theorem and one of the Conley-Zehnder theorem. In both cases the periodic orbits are of a most simple as possible braid type. Then he establishes an equivariant version of the Brouwer translation theorem enabling new proofs of some results concerning the rotation set and periodic orbits of diffeomorphisms of the circle.

In the first chapter the author deals with the theory of monotone twist maps. At the beginning he discusses several examples in order to motivate and illustrate results and central ideas of the theory. After recalling basic facts and definitions he explains the Aubry-Mather theory. This is a variational approach appropriate especially for the conservative case leading to criteria for the existence of periodic orbits. Then the author focuses his interest on the Birkhoff theory which is suitable for both conservative as well as dissipative systems. He applies this topological approach in order to obtain a precise description of the dynamics on invariant curves. After representing connections to the KAM-theory he studies area decreasing maps having so-called Birkhoff attractors.

The second chapter bases on the fact that each diffeomorphism of the closed annulus isotopic to the identity can be written as a composition of some monotone twist maps. Hence the author uses generalizations of some of the methods described in the first chapter for studying this kind of diffeomorphisms. Firstly, he proves a version of the Poincaré- Birkhoff theorem and one of the Conley-Zehnder theorem. In both cases the periodic orbits are of a most simple as possible braid type. Then he establishes an equivariant version of the Brouwer translation theorem enabling new proofs of some results concerning the rotation set and periodic orbits of diffeomorphisms of the circle.

Reviewer: H.Kriete (Aachen)

##### MSC:

37B99 | Topological dynamics |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

54H20 | Topological dynamics (MSC2010) |

39B12 | Iteration theory, iterative and composite equations |