On the minimum positive entropy for cycles on trees

Abstract:

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