Difféomorphismes de Smale des surfaces / C. Bonatti et R. Langevin ; avec la collaboration de E. Jeandenans

This work is organized as follows. It begins with an interesting introduction on the theory of dynamical systems, with special attention to the problem of classification of hyperbolic dynamics, and then a brief presentation of the main results of this work. Chapter 1 is devoted to the study of satured hyperbolic sets of the Smale diffeomorphisms (i.e. hyperbolic sets which are equal to the intersection of their invariant manifolds), which are a generalization of the basic sets of the theory of Smale of hyperbolic dynamics, and also of the invariant neighbourhoods. In Chapter 2 the invariant curves are studied. Canonical (up to conjugacy) invariant neighbourhoods of the satured hyperbolic sets (called domains) are studied in Chapter 3 and it is shown that a Smale diffeomorphism can be reconstructed from its restrictions to the domains, by gluing the domains along their boundary. The construction of Markov partitions is considered in Chapter 4. So in Chapter 5, geometrical Markov partitions and topological conjugacy of Smale diffeomorphisms, it is proved that the dynamics restricted to the domains is characterized by the geometrical type of some Markov partition of the satured hyperbolic set: it is a simple combinatorics describing in which order, position and direction the image of some rectangle of the Markov partition crosses the rectangles. As a consequence of the previous study, in Chapter 6 it is observed that the dynamics of a Smale diffeomorphism on a domain is characterized by the pattern of the invariant curves. The last two chapters are authored by the first author and E. Jeandenans. Some of the abstract geometrical types do not correspond to any Smale diffeomorphism on compact surfaces, so in Chapter 7, they define the genus of a type, as a minorant of the genus of any compact surface on which the type can be realized as the geometrical type of a Markov partition of some satured hyperbolic set; then they characterize the geometrical types of finite genus. Finally, Chapter 8 is devoted to the study of basic sets and pseudo-Anosov homeomorphisms. (Zentralblatt)

Bibliogr. p. 231-235