Wandering continua for rational maps

TL;DR

This paper characterizes when Lattès maps have wandering continua, showing that only flexible ones admit such, with these continua being line segments along bi-infinite geodesics.

Contribution

It provides a precise criterion linking flexibility of Lattès maps to the existence of wandering continua, describing their geometric nature.

Findings

Flexible Lattès maps admit wandering continua.

Wandering continua are line segments in bi-infinite geodesics.

Only flexible Lattès maps have such wandering continua.

Abstract

We prove that a Latt' es map admits an eventually simply-connected wandering continuum precisely when it is flexible. The simply-connected wandering continuum is a line segment in a bi-infinite geodesic under the flat metric.