On the growth rate inequality for periodic points in the two sphere

Abstract

Let $f:S^2\to S^2$ be a continuous map such that $degf=d,|d|>1$. Suppose $f$ has two attracting fixed points denoted $N$ and $S$ and let $A=S^2\setminus \{N,S\}$. Assume that if a loop ${\hat{I}}^3\subset f^{-1}(A)$ is homotopically trivial in $A$, then $f({\hat{I}}^3)$ is also homotopically trivial in $A$. Then, for all $n$, $f$ has at least $|d^n-1|$ fixed points.