Let \(f:S^2 \rightarrow S^2\) be a map of degree \(d\), \(\vert d \vert >1\), and let \(N_n f\) denote the number of fixed points of \(f^n\). The following question is considered: when does the growth rate inequality \(\textrm{lim}\, \sup\, \frac{1}{n} \textrm{log}\, N_n f \geq \textrm{log}\, \vert d \vert \) holds for \(f\)? A special instance of this problem was first considered by M. Shub [in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS). 99–120 (2006; Zbl 1113.37006), Problem 3]. If the growth rate inequality holds for \(f\), it is said that \(f\) has the rate.
This paper considers the case when \(f\) is a degree-\(d\) branched covering, that is, \(f\) is a local homeomorphism except at a finite critical set \(C\), and it is proved that in certain cases \(f\) has the rate. Suppose first that there exists a completely invariant simply connected region \(R\) whose boundary is locally connected, and that there does not exist only one critical point in the boundary of \(R\) that has multiplicity \(d-1\) and is fixed by \(f\) (if there exist such a critical point, then it is known that \(f\) may not have the rate). The main result shows that if \(f\) satisfies these conditions, then \(f\) has the rate. It is also shown that if there exists a simply connected open set \(U\) whose closure is disjoint from the set of critical values and such that \(\overline {f^{-1}(U)} \subset U\), then \(f\) has the rate.