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

Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\text{Pos}_n$, $\text{Irr}_n$ and $\text{Pos}_n\setminus\text{Irr}_n$. 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 $\log(\lambda_n)$. We prove that this entropy is minimum in $\text{Pos}_n$. Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ also minimizes the entropy in the family $\text{Irr}_n$. We also prove that the minimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (which is nonempty only for composite integers $n\ge6$) is $\log(\lambda_{n/p})/p$, where $p$ is the least prime factor of $n$.