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

TL;DR

This paper investigates the growth rate of fixed points for certain self-maps of spheres, establishing conditions under which the number of fixed points grows exponentially with respect to the map's degree.

Contribution

It introduces a new inequality relating the degrees of self-maps of spheres and their fixed point growth, under specific inverse image conditions.

Findings

Fixed points grow exponentially with base |d| > 1

Established a growth rate inequality for self-maps of spheres

Connected fixed point growth to the degrees of the map and its restriction

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