The minimum tree for a given zero‐entropy period

Abstract

We answer the following question: given any n ∈ ℕ, which is the minimum number of endpoints en of a tree admitting a zero‐entropy map f with a periodic orbit of period n? We prove that , where n = s1s2 … sk is the decomposition of n into a product of primes such that si ≤ si+1 for 1 ≤ i < k. As a corollary, we get a criterion to decide whether a map f defined on a tree with e endpoints has positive entropy: if f has a periodic orbit of period m with em > e, then the topological entropy of f is positive.