On the set of periods for $\sigma$ maps

Abstract:

Let $\sigma$ be the topological graph shaped like the letter $\sigma$. We denote by $0$ the unique branching point of $\sigma$, and by ${\mathbf {O}}$ and ${\mathbf {I}}$ the closures of the components of $\sigma \backslash \{ 0\}$ homeomorphics to the circle and the interval, respectively. A continuous map from $\sigma$ into itself satisfying that $f$ has a fixed point in ${\mathbf {O}}$, or $f$ has a fixed point and $f(0) \in {\mathbf {I}}$ is called a $\sigma$ map. These are the continuous self-maps of $\sigma$ whose sets of periods can be studied without the notion of rotation interval. We characterize the sets of periods of all $\sigma$ maps.