Forward triplets and topological entropy on trees

Abstract

<p style='text-indent:20px;'>We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> has positive entropy if and only if some iterate <inline-formula><tex-math id="M2">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> has a periodic orbit with three aligned points consecutive in time, that is, a triplet <inline-formula><tex-math id="M3">\begin{document}$ (a,b,c) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M4">\begin{document}$ f^k(a) = b $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ f^k(b) = c $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> belongs to the interior of the unique interval connecting <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ c $\end{document}</tex-math></inline-formula> (a <i>forward triplet</i> of <inline-formula><tex-math id="M9">\begin{document}$ f^k $\end{document}</tex-math></inline-formula>). We also prove a new criterion of entropy zero for simplicial <inline-formula><tex-math id="M10">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M11">\begin{document}$ P $\end{document}</tex-math></inline-formula> based on the non existence of forward triplets of <inline-formula><tex-math id="M12">\begin{document}$ f^k $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M13">\begin{document}$ 1\le k&lt;n $\end{document}</tex-math></inline-formula> inside <inline-formula><tex-math id="M14">\begin{document}$ P $\end{document}</tex-math></inline-formula>. Finally, we study the set <inline-formula><tex-math id="M15">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> of all <inline-formula><tex-math id="M16">\begin{document}$ n $\end{document}</tex-math></inline-formula>-periodic patterns <inline-formula><tex-math id="M17">\begin{document}$ P $\end{document}</tex-math></inline-formula> that have a forward triplet inside <inline-formula><tex-math id="M18">\begin{document}$ P $\end{document}</tex-math></inline-formula>. For any <inline-formula><tex-math id="M19">\begin{document}$ n $\end{document}</tex-math></inline-formula>, we define a pattern that attains the minimum entropy in <inline-formula><tex-math id="M20">\begin{document}$ \mathcal{X}_n $\end{document}</tex-math></inline-formula> and prove that this entropy is the unique real root in <inline-formula><tex-math id="M21">\begin{document}$ (1,\infty) $\end{document}</tex-math></inline-formula> of the polynomial <inline-formula><tex-math id="M22">\begin{document}$ x^n-2x-1 $\end{document}</tex-math></inline-formula>.</p>