On the minimum positive entropy for cycles on trees

Abstract

Abstract. 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 x n − 2x − 1 in (1, +∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(λn). For n = m k , 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.

Forward triplets and topological entropy on trees

“…) is a well defined pattern, which we call a π-reduced (or simply reduced ) pattern of P. The following result (Proposition 9.5 of [4]) summarizes some properties of a reduced pattern for the specific case of periodic patterns. Proposition 1.…”

“…Figure 2. An interval model (T, P, f ) and the corresponding pattern, that can be identified with the permutation (3,4,2,5,1). that two pointed trees (T, P ) and (S, Q) are equivalent if there exists a bijection φ : P −→ Q which preserves discrete components.…”

“…The third main result of this paper gives in fact a lower bound for its entropy. In [4], the authors considered the problem of determining, for each n ∈ N, the minimum entropy in the set Pos n of all n-periodic patterns with positive entropy. Of course every periodic pattern of period 1 or 2 has entropy zero, so the problem makes sense only for n ≥ 3.…”

“…In [4] it is proved that h(Q n ) = log(λ n ), where λ n is the unique real root in (1, ∞) of the polynomial x n − 2x − 1. It can be seen that the sequence (λ n ) ∞ n=1 is decreasing and tends to 1 as n → ∞.…”

“…It can be seen that the sequence (λ n ) ∞ n=1 is decreasing and tends to 1 as n → ∞. The main theorem in [4] states that, when n has the particular form n = p k for a prime p and k ≥ 1, indeed the pattern Q n minimizes the entropy in Pos n . In consequence, h(P) ≥ log(λ n ) for any pattern P ∈ Pos n .…”