The growth rate inequality for Thurston maps with non-hyperbolic orbifolds

Let f : S 2 → S 2 be a continuous map of degree d, |d| > 1, and let Nnf 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 lim sup 1 n log Nnf ≥ log |d| holds for f or f has exactly two critical points which are fixed and totally invariant.