Rotation set for maps of degree 1 on the graph sigma

TL;DR

This paper studies rotation sets for degree 1 maps on a specific graph structure called sigma, proving the set is closed with finitely many components and contains periodic points for all rational rotation numbers.

Contribution

It establishes that for degree 1 maps on the graph sigma, the rotation set is closed, finitely connected, and contains periodic points for all rational rotation numbers.

Findings

Rotation set is closed for degree 1 maps on sigma.

Rotation set has finitely many connected components.

Periodic points exist for all rational rotation numbers in the set.

Abstract

For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e. the set of all rotation numbers) may not be connected and it is not known in general whether it is closed. The graph sigma is the space consisting in an interval attached by one of its endpoints to a circle. We show that, for a map of degree 1 on the graph sigma, the rotation set is closed and has finitely many connected components. Moreover, for all rational numbers r in the rotation set, there exists a periodic point of rotation number r.