Periodic orbits of large diameter for circle maps

TL;DR

This paper investigates how the existence of a large periodic orbit in a circle map's lift influences its set of periods, establishing that for degree d ≥ 1, a large orbit implies all periods are present.

Contribution

It proves that large periodic orbits in degree d ≥ 1 circle maps guarantee the existence of all periods, extending understanding of orbit structure in such maps.

Findings

Large orbits imply all periods for degree d ≥ 1.

Counterexamples show the result fails for non-positive degree.

Provides conditions linking orbit size to period set completeness.

Abstract

Let $f$ be a continuous circle map and let $F$ be a lifting of $f$. In this note we study how the existence of a large orbit for $F$ affects its set of periods. More precisely, we show that, if $F$ is of degree $d\geq 1$ and has a periodic orbit of diameter larger than 1, then $F$ has periodic points of period $n$ for all integers $n\geq 1$, and thus so has $f$. We also give examples showing that this result does not hold when the degree is non positive.