Dynamics of annulus coverings II: periodic points

TL;DR

This paper investigates the dynamics of covering maps of the annulus, showing that such maps preserve the number of periodic points in each period compared to the map $z^d$, under certain conditions.

Contribution

It establishes a lower bound on the number of periodic points for annulus covering maps that preserve an essential compact set, extending known results for the map $z^d$.

Findings

At least as many periodic points as $z^d$ in each period.

Periodic points are preserved under annulus coverings with degree greater than one.

The result applies to maps preserving essential compact subsets of the annulus.

Abstract

Let $f$ be a covering map of the open annulus $A= S^1\times (0,1)$ of degree $d$ , $|d|>1$. Assume that $f$ preserves an essential (i.e not contained in a disk of $A$) compact subset $K$. We show that $f$ has at least the same number of periodic points in each period as the map $z^d$ in $S^1.$