On the set of periods of sigma maps of degree 1

We study the set of periods of degree 1 continuous maps from σ into itself, where σ denotes the space shaped like the letter σ (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 σ-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 σ; 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.