Elementary proof of the continuity of the topological entropy at $heta=underline{1001}$ in the Milnor-Thurston world

摘要

In 1965, Adler, Konheim and McAndrew introduced the topological entropy of a given dynamical system, which consists of a real number that explains part of the complexity of the dynamics of the system. In this context, a good question could be if the topological entropy $H_{mathrm{top}} (f)$ changes continuously with $f$. For continuous maps this problem was studied by Misisurewicz, Slenk and Urbański. Recently, and related with the lexicographic and the Milnor-Thurston worlds, this problem was studied by Labarca and others. In this paper we will prove, by elementary methods, the continuity of the topological entropy in a maximal periodic orbit ($ heta= underline{1001} $) in the Milnor-Thurston world. Moreover, by using dynamical methods, we obtain interesting relations and results concerning the largest eigenvalue of a sequence of square matrices whose lengths grow up to infinity.