On the Minimum Positive Entropy for Cycles on Trees

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