Homotopic rotation sets for higher genus surfaces

This paper states a definition of homotopic rotation set for higher genus surface homeomorphisms, as well as a collection of results that justify this definition. We first prove elementary results: we prove that this rotation set is star-shaped, we discuss the realisation of rotation vectors by orbits or periodic orbits and we prove the creation of new rotation vectors for some configurations. Then we use the theory developped by Le Calvez and Tal (“Forcing theory for transverse trajectories of surface homeomorphisms,” <italic>Invent. Math.</italic> (212), 2018) to obtain two deeper results: If the homotopical rotation set contains the direction of a closed geodesic which has a self-intersection, then there exists a rotational horseshoe and hence infinitely many periodic orbits in many directions. If the homotopical rotation set contains the directions of two closed geodesics that meet, there exists infinitely many periodic orbits in many directions.