One of the main dynamical invariants related to a surface homeomorphism isotopic to identity is its rotation set, which describes the asymptotic average speeds and directions with which the points “rotate” around the surface under the action of the homeomorphism. In the case of a torus homeomorphism, in particular, many results link the shape or the size of the rotation set to dynamical properties of the homeomorphism.

The aim of this work is to generalize to the case of surfaces of genus \(\geq 2\) a certain number of results, well-known on the torus, for homeomorphisms with a “big” rotation set: positivity of the entropy, realization of rotation vectors by periodic points, bounded deviations, etc. The leading tool used is the forcing theory by Le Calvez and Tal, based on the construction of a transverse foliation and the study of trajectories of points relatively to this foliation.

The first two chapters present some preliminary results in this general context. In chapter 3, the author conducts a general study on the asymptotic cycles of points whose trajectories have homological directions that intersect. He shows that this situation is sufficient to ensure the positivity of the entropy, which leads him to derive a generalization of two well-known results on the torus, Llibre-Mackay and Franks theorems.

The author obtains in the case of a surface of genus \(\geq 2\) that a homeomorphism with a “big” rotation set has positive entropy, and the author manages to realize many rational points of the rotation set as rotation vectors of periodic points. Finally, in chapter 4, the author uses this last result to show that a homeomorphism for which 0 lies in the interior of the rotation set has bounded deviations, generalizing again a well-known property on the torus. He concludes with some consequences of this result.