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

Abstract

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.

Figures (18)

• Figure 1: Three pointed trees that should be equivalent.
• Figure 2: Two pointed trees that should be equivalent.
• Figure 3: Letf and center: two periodic models exhibiting the same pattern (an arrow from $a$ to $b$ codifies that the image of $a$ by the corresponding map is $b$). Right: a possible representation of the pattern exhibited by both models.
• Figure 4: A representation of the 6-periodic pattern in Figure \ref{['exemples3']}(right).
• Figure 5: Left: a 7-periodic pattern $\mathcal{P}$. Center: the canonical model of $\mathcal{P}$. Right: the points of $\mathcal{P}$ embedded on a 4-star.

Theorems & Definitions (39)

• Remark 1.1: Standing assumption
• Proposition 1.2
• Theorem 1.3
• Proposition 2.1
• Remark 2.2: Standing convention
• Proposition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 1
• Proposition 2