Zero entropy cycles on trees: from Topology to Combinatorics and an application to star maps

In this paper we give a fully combinatorial description of the zero entropy periodic patterns on trees. Unlike previously known characterizations of such patterns, our criterion is independent of any particular topological realization of the pattern and provides, thus, a practical and fast algorithm to test zero entropy. As an application, consider a $k$-star $T$ (a tree with $k$ edges attached at a unique branching point of valence $k$) and the set $\mathcal{F}_{n,k}$ of all continuous maps $\map{f}{T}$ having a periodic orbit of period $n$ properly contained in $T$ (each edge of $T$ contains at least one point of the orbit). We find all pairs $(n,k)$ such that $\mathcal{F}_{n,k}$ contains maps of entropy zero, and we describe the patterns of such zero-entropy orbits.