Rotation sets for random compositions of $\T$ homeomorphisms
Catalina Freijo, Fabio Tal
TL;DR
This work extends rotation theory to random compositions of torus homeomorphisms by formulating cocycles over shift spaces and introducing multiple rotation-set notions for these cocycles. It establishes foundational properties: the Misiurewicz-Ziemian-type rotation set $\rho_{mz}(\widetilde{F})$ is nonempty, compact, and connected, and it satisfies natural inclusions $\rho_{point}(\widetilde{F})\subset \rho_{mz}(\widetilde{F})\subset \rho_{mes}(\widetilde{F})$, with connections to invariant measures and periodic words. The paper also analyzes continuity (showing semi-continuity under weak-* convergence), essential/inessential point structure, and the shape of rotation sets in special cases, proving convexity for some ε-pseudo-orbit constructions and outlining several open questions about convexity and measure-approximation in the general setting. Overall, it provides a framework and initial results for the rotational behavior of random torus dynamics, paving the way for further structural and geometrical insights in random dynamical systems on the torus.
Abstract
We study cocycles of homeomorphisms of $\T$ in the isotopy class of the identity over shift spaces, using as a tool a novel definition of rotation sets inspired in the classical work of Miziurewicz and Zieman. We discuss different notions of rotation sets, for the full cocyle as well as for measures invariant by the shift dynamics on the base. We present some initial results on the shape of rotation sets, continuity of rotation sets for shift-invariant measures, and bounded displacements for irrotational cocyles, as well as a few interesting examples in an attempt to motive the development of the topic.