Forward triplets and topological entropy on trees

Abstract

We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map $f$ has positive entropy if and only if some iterate $f^k$ has a periodic orbit with three aligned points consecutive in time, that is, a triplet $(a,b,c)$ such that $f^k(a)=b$, $f^k(b)=c$ and $b$ belongs to the interior of the unique interval connecting $a$ and $c$ (a \emph{forward triplet} of $f^k$). We also prove a new criterion of entropy zero for simplicial $n$-periodic patterns $P$ based on the non existence of forward triplets of $f^k$ for any $1\le k<n$ inside $P$. Finally, we study the set $\mathcal{X}_n$ of all $n$-periodic patterns $P$ that have a forward triplet inside $P$. For any $n$, we define a pattern that attains the minimum entropy in $\mathcal{X}_n$ and prove that this entropy is the unique real root in $(1,\infty)$ of the polynomial $x^n-2x-1$