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

TL;DR

This paper establishes a lower bound on the number of fixed points for certain continuous maps on the 2-sphere with degree greater than one, under specific homotopy conditions and the presence of attracting fixed points.

Contribution

It proves a growth rate inequality for periodic points of maps on the 2-sphere with degree greater than one, under particular topological and homotopy assumptions.

Findings

At least |d^n - 1| fixed points for all n

Conditions on homotopy triviality influence fixed point count

Results extend understanding of periodic points in sphere maps

Abstract

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