On the set of periods of sigma maps of degree 1

TL;DR

This paper investigates the set of periods for degree 1 continuous maps on a sigma-shaped space, revealing conditions under which all periods occur and how the structure influences periodic point sets.

Contribution

It characterizes the set of periods for sigma maps with degree 1, linking rotation intervals to periodic point sets and constructing examples with combined period sets.

Findings

When the rotation interval contains an integer, all periods except possibly 1 or 2 occur.

Constructed examples show the set of periods can combine those of circle and 3-star maps.

Existence of a large periodic orbit outside the circuit implies all periods are present.

Abstract

We study the set of periods of degree 1 continuous maps from sigma into itself, where sigma denotes the space shaped like the letter sigma (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration have degree 1, the rotation theory can be used. We show that, when the interior of the rotation interval contains an integer, then the set of periods (of periodic points of any rotation number) is the set of all integers except maybe 1 or 2. We exhibit degree 1 sigma-maps f whose set of periods is a combination of the set of periods of a degree 1 circle map and the set of periods of a 3-star (that is, a space shaped like the letter Y). Moreover, we study the set of periods forced by periodic orbits that do not intersect the circuit of sigma; in particular, when there exists such a periodic orbit whose diameter (in the covering space) is at least 1, then there exist periodic points of all periods.