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

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