Invariant Peano curves of expanding Thurston maps

TL;DR

This paper proves that expanding Thurston maps have high iterates semi-conjugate to power maps on the circle, and constructs invariant Peano curves that intertwine these maps, revealing a topological structure akin to invariant curves.

Contribution

It introduces a method to construct invariant Peano curves for expanding Thurston maps, linking their dynamics to simple power maps on the circle.

Findings

High iterates of expanding Thurston maps are semi-conjugate to $z^d$ on $S^1$

Existence of invariant Peano curves for these maps

Provides a topological model for the dynamics of expanding Thurston maps

Abstract

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma\colon S^1\to S^2$ (onto), such that $F\circ \gamma(z) = \gamma(z^d)$ (for all $z\in S^1$).