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

Abstract Consider, for any integer $n\ge 3$ , the set $\operatorname {\mathrm {Pos}}_n$ of all n -periodic tree patterns with positive topological entropy and the set $\operatorname {\mathrm {Irr}}_n\subset \operatorname {\mathrm {Pos}}_n$ of all n -periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\operatorname {\mathrm {Pos}}_n$ , $\operatorname {\mathrm {Irr}}_n$ and $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$ . Let $\unicode{x3bb} _n$ 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 $\log (\unicode{x3bb} _n)$ . We prove that this entropy is minimum in $\operatorname {\mathrm {Pos}}_n$ . Since the pattern $\mathcal {Q}_n$ is irreducible, $\mathcal {Q}_n$ also minimizes the entropy in the family $\operatorname {\mathrm {Irr}}_n$ . We also prove that the minimum positive entropy in the set $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$ (which is non-empty only for composite integers $n\ge 6$ ) is $\log (\unicode{x3bb} _{n/p})/p$ , where p is the least prime factor of n .