On the minimum positive entropy for cycles on trees

Abstract

Consider, for any n ∈ N, the set Pos n of all n-periodic tree patterns with positive topological entropy and the set Irr n ⊊ Pos n of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Pos n and Irr n . Let λ n be the unique real root of the polynomial x n − 2x − 1 in (1, + ∞). We explicitly construct an irreducible n-periodic tree pattern Q n whose entropy is log(λ n ). For n = m k , where m is a prime, we prove that this entropy is minimum in the set Pos n . Since the pattern Q n is irreducible, Q n also minimizes the entropy in the family Irr n