  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 $d =
\mathrm{deg}(f)/\mathrm{deg}(f_{|P}) \in \mathbb Z$.
