A rotational hyperbolic theory for surface homeomorphisms

TL;DR

This paper introduces a rotational hyperbolic framework for surface homeomorphisms, using ergodic measures and homoclinic classes to analyze rotational behavior and connections, supported by forcing theory.

Contribution

It develops a rotational hyperbolic theory for surface homeomorphisms, defining rotational homoclinic classes and characterizing heteroclinic connections, extending forcing theory techniques.

Findings

Network of horseshoes representing rotational behavior

Five characterizations of heteroclinic connections

Creation of periodic points with homotopically bounded deviations

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].