Sphere branched coverings and the growth rate inequality
  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. 
J. Iglesias, A.Portela, A. Rovella and J. Xavier
math.DS