The rotation set around a fixed point for surface homeomorphisms (Q2840584)

In this lovely little book the author deals with local topological dynamics in dimension two, i.e. iterations of a germ of a surface homeomorphism nearby a fixed point.NEWLINENEWLINEThe main point of view used to analyze these iterations is that of the rotation set, which describes the speed with which the points nearby the fixed point rotate around it. Among the techniques used, the transverse foliations, which were recently introduced by Patrice Le Calvez, play a key role.NEWLINENEWLINEAlthough the concept of rotation number has a long and distinguished history going back to Henry Poincaré, the author provides for the first time an explicit general definition of rotation sets for germs of surface homeomorphisms nearby fixed points and starts a systematic study of their properties.NEWLINENEWLINEThe book is structured in the following way.NEWLINENEWLINEIn the first chapter the author recalls the definition of rotation number for a circle and that of rotation set for a compact annulus, and then he provides an extension of this definition for the open annulus. This is particularly useful for nonspecialists interested to get the main gist of the subject and to prepare them for the subsequent chapters.NEWLINENEWLINEIn the second chapter, the author presents the definition and proves some properties for two invariants, the set of local rotation and the interval of local rotation. Among the new results developed in this chapter, let me mention the following two:NEWLINENEWLINE1) The possibility to define an interval of local rotation via the dynamics on curves going through the fixed point under consideration. 2)The possibility of blowing up a fixed point and replacing it with a circle (à la Gambaudo-Le Calvez-Pécou) is equivalent to the differentiability at this point, up to topological change of coordinates. Moreover, these properties are implied by the existence of a germ of a curve which is disjoint from all its iterates. (The author points out that these results have been obtained in collaboration with François Béguin and Sylvain Crovisier.)NEWLINENEWLINEThe third chapter deals with what happens when the rotation set is neither empty nor reduced to \(\{0\}\), while the fourth chapter analyzes the situation in the latter two cases, focusing in particular on the study of isolated fixed points having Poincaré-Lefschetz index different from one. Some highlights of new results presented in these two chapters are the following:NEWLINENEWLINE1) The interval of local rotation coincides with the convex envelope of the set of local rotation.NEWLINENEWLINE 2) The set of local rotation can be used to detect certain periodic orbits.NEWLINENEWLINE 3) The local transversality of a foliation and of an isotopy is an open property. This allows one to perturb a transverse foliation to obtain a simpler one in a topological sense. This is related to the equivariant foliation theorem of Patrice Le Calvez.NEWLINENEWLINE Using these tools the author revisits old results, providing new proofs that sometimes strengthen the statements of the previously known results, like in the case of the three-four fixed points theorem of Matsumoto and in the study of isolated fixed points of index different from one. In particular the author proves the existence of transversally hyperbolic sectors where one can define a ``good'' Lyapunov function.NEWLINENEWLINEThe text concludes with an epilogue giving a proof of a theorem on global dynamics (Matsumoto theorem on the three-four fixed points), showing how the objects introduced in the previous chapters can be used to strengthen known results, and with two appendices mostly presenting an introduction to local dynamics of foliations.NEWLINENEWLINEThis clear, richly illustrated and lively written work contains both known results of local dynamics, revisited from the point of view of the rotation set, and new interesting theorems and ingenious constructions. It is well suited both for beginners interested in local topological dynamics in dimension two as well as specialists.