Characterization of the tree cycles with minimum positive entropy for any period

Abstract

Consider, for any integer n ≥ 3, 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, Irrn and Posn \ 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). We prove that this entropy is minimum in Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn. We also prove that the minimum positive entropy in the set Posn \ Irrn (which is non-empty only for composite integers n ≥ 6) is log(λn/p)/p, where p is the least prime factor of n