On the growth rate inequality for self-maps of the sphere

Let $S^m=\{x_0^2+x_1^2+\cdots +x_m^2=1\}$ and $P=\{x_0=x_1=0\}\cap S^m$. Suppose that $f$ is a self–map of $S^m$ such that $f^{-1}(P)=P$ and $\mathrm{\mid }\mathrm{d}\mathrm{e}\mathrm{g}(f_{\mathrm{\mid }P})\mathrm{\mid }<\mathrm{\mid }\mathrm{d}\mathrm{e}\mathrm{g}(f)\mathrm{\mid }$. Then, the number of fixed points of $f^n$ grows at least exponentially with base $\mathrm{\mid }d\mathrm{\mid }>1$, where $d=\mathrm{d}\mathrm{e}\mathrm{g}(f)\mathrm{/}\mathrm{d}\mathrm{e}\mathrm{g}(f_{\mathrm{\mid }P})\in \mathbb{Z}$.