Sphere branched coverings and the growth rate inequality

Abstract

We show that the growth inequality rate $$\mathrm{limsup}\frac{1}{n}\log (\#Fix(f^n))\ge \log d$$ 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.