Projective transformations of rotation sets

Abstract

We give a new proof and extend a result of J. Kwapisz: whenever a set C is realized as the rotation set of some torus homeomorphism, the image of C under certain projective transformations is also realized as a rotations set.The concept of rotation set, introduced by M. Misiurewicz and K. Ziemian in [5], is one of the most important tools to study the global dynamics of homeomorphisms of the torus T 2 . If f is a homeomorphism of T 2 isotopic to the identity, and F is a lift of f to R 2 , the rotation set of F is a compact convex subset of the plane which describes "at what speeds and in what directions the orbits of f rotate around the torus". One of the main problems in the theory is to determine which compact convex subsets of R 2 can be realized as the rotations sets of some torus homeomorphisms. For compact convex subsets with empty interiors (i.e. singletons and segments), a conjectural answer to the problem has been formulated by J. Franks and M. Misiurewicz (see [1]). Fifteen years ago, J. Kwapicz has introduced a technical tool which allows to simplify the problem. Namely, he observed that, if a compact convex set C ⊂ R 2 is realized as the rotation of a certain torus diffeomorphism, and if a projective transformation L maps C to a bounded set of the plane, then L(C) can be realized as the rotation of another torus diffeomorphism (see [2, section 2]).Kwapisz's proof requires to consider the suspension of the initial torus homeomorphism, and to apply a theorem of D. Fried to find a new surface of section for this flow, in the appropriate cohomology class. Fried's theorem works only for C 1 flows; this forces Kwapisz to consider only rotation sets of C 1 diffeomorphisms, whereas the natural setting for his result would be rotation sets of homeomorphisms. The purpose of the present note is to provide a more elementary proof of Kwapisz's result. Our proof remains at the level of surfaces homeomorphisms, i.e. does not require to consider a flow on a three-dimensional manifold. It does not make use of Fried's theorem (in some sense, we replace it by the more classical fact that the only surface with fundamental group isomorphic to Z 2 is the torus T 2 ). As a consequence, it works for surface homeomorphisms without any differentiability assumption. This might be of interest in relation with some recent works related to the Franks-Misiurewicz conjecture (see [4,3], and the example of Avila quoted in these papers). * S.C was partially supported by the ERC project 692925 NUHGD.