Forward triplets and topological entropy on trees

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