The growth rate inequality for Thurston maps with non hyperbolic orbifolds

TL;DR

This paper investigates the growth rate of fixed points for Thurston maps with non-hyperbolic orbifolds, establishing conditions under which the growth rate inequality holds or the map has a specific critical point structure.

Contribution

It proves a dichotomy for Thurston maps with non-hyperbolic orbifolds regarding fixed point growth or critical point invariance.

Findings

Growth rate inequality holds unless the map has two fixed, totally invariant critical points.

Identifies a special case with exactly two fixed, totally invariant critical points.

Provides conditions linking orbifold type to fixed point dynamics.

Abstract

Let $f: S^2 \to S^2$ be a continuous map of degree $d$, $|d|>1$, and let $N_nf$ denote the number of fixed points of $f^n$. We show that if $f$ is a Thurston map with non hyperbolic orbifold, then either the growth rate inequality $\limsup \frac{1}{n} \log N_nf\geq \log |d|$ holds for $f$ or $f$ has exactly two critical points which are fixed and totally invariant.