The growth rate inequality for Thurston maps with non hyperbolic orbifolds

Abstract

Let $f:S^2\to S^2$ be a continuous map of degree $d$, $|d|>1$, and let $N_{n}f$ 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 $\mathrm{limsup}\frac{1}{n}\log N_{n}f\ge \log |d|$ holds for $f$ or $f$ has exactly two critical points which are fixed and totally invariant.