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

Abstract

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 $|\mathrm{deg}(f_{|P})|<|\mathrm{deg}(f)|$. Then, the number of fixed points of $f^n$ grows at least exponentially with base $|d|>1$, where $d=\mathrm{deg}(f)/\mathrm{deg}(f_{|P})\in \mathbb{Z}$.

10 pages, 1 figure