Sphere branched coverings and the growth rate inequality

TL;DR

This paper proves that for certain branched coverings of the sphere with a specific invariant region, the growth rate of fixed points under iteration is at least the logarithm of the degree, except in some special cases.

Contribution

It establishes the growth inequality for a class of sphere branched coverings with invariant regions, extending previous results and identifying exceptions.

Findings

Growth inequality holds for most branched coverings with invariant regions.

Exceptions are identified with known counterexamples.

Provides conditions under which the inequality is valid.

Abstract

We show that the growth inequality rate $$\limsup \frac{1}{n} \log (\# Fix (f^n))\geq \log d$$ holds for branched coverings of degree $d$ of the sphere $S^2$ having a completely invariant simply connected region $R$ with locally connected boundary, except in some degenerate cases with known couterexamples.