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

Consider, for any integer $n\ge 3$, 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 $(1,+\mathrm{\infty })$. 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 $n\ge 6$) is $\log ({\lambda }_{n\mathrm{/}p})\mathrm{/}p$, where $p$ is the least prime factor of $n$.