Rotation sets and actions on curves

TL;DR

This paper explores the complex dynamics of surface homeomorphisms acting on a fine curve graph, revealing richer behaviors than classical cases, especially on tori, with implications for understanding rotation sets and translation lengths.

Contribution

It introduces the fine curve graph for surfaces, analyzes the full spectrum of isometries induced by homeomorphisms, and connects these dynamics to rotation sets on tori, providing new characterizations.

Findings

Homeomorphisms induce parabolic, elliptic, and hyperbolic isometries.

All positive real numbers can be realized as asymptotic translation lengths.

Hyperbolic actions relate to the area of rotation sets on tori.

Abstract

We study the action of the homeomorphism group of a surface $S$ on the fine curve graph ${\mathcal C }^\dagger(S)$. While the definition of $\mathcal{C}^\dagger(S)$ parallels the classical curve graph for mapping class groups, we show that the dynamics of the action of ${\mathrm{Homeo}}(S)$ on $\mathcal{C}^\dagger(S)$ is much richer: homeomorphisms induce parabolic isometries in addition to elliptics and hyperbolics, and all positive reals are realized as asymptotic translation lengths. When the surface $S$ is a torus, we relate the dynamics of the action of a homeomorphism on $\mathcal{C}^\dagger(S)$ to the dynamics of its action on the torus via the classical theory of rotation sets. We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.