Ping-pong dynamics of hyperbolic-like actions with non-simple points

TL;DR

This paper studies the dynamics of hyperbolic-like groups acting on the circle, providing explicit constructions of ping-pong partitions for certain stabilizers, advancing understanding of their geometric properties.

Contribution

It explicitly constructs ping-pong partitions for hyperbolic-like groups with non-cyclic point stabilizers, confirming conjectural dynamics behaviors.

Findings

Existence of proper ping-pong partitions for non-cyclic stabilizers

Explicit construction of ping-pong partitions

Advancement in understanding hyperbolic-like group actions

Abstract

A hyperbolic-like group is a subgroup of $\operatorname{Homeo}_+(S^1)$ such that every non-trivial element has exactly two fixed points, one attracting and one repelling. We investigate the ping-pong dynamics of hyperbolic-like groups, inspired by a conjecture of Bonatti. We show the existence of a proper ping-pong partition for any pair of non-cyclic point stabilizers. More precisely, our results explicitly provide such a ping-pong partition.