A rotational hyperbolic theory for surface homeomorphisms

Abstract

We develop a rotational hyperbolic theory for surface homeomorphisms. We use the equivalence relation on ergodic measures that have nontrivial rotational behaviour defined in [arXiv:2312.06249] to define a rotational counterpart of homoclinic classes. These allows to produce a network of horseshoes representing the whole rotational behaviour f the homeomorphism. We also study the counterpart of heteroclinic connections and give 5 different characterizations of such connections. The main technical tool is the forcing theory of Le Calvez and Tal [arXiv:1503.09127], [arXiv:1803.04557], and in particular a result of creation of periodic points that can also be seen as a statement of homotopically bounded deviations [arXiv:2511.14222]. This theoretical article is followed by a paper focused of some applications of it to the case of homeomorphisms with big rotation set [arXiv:2511.15220].