The entropy of multiplicative subshifts on trees

Abstract

The aim of this paper is to study the topological entropy of a q-multiplicative subshift on tree (q-MST), that is, every boundary of the tree follows the rule of some one-dimensional multiplicative integer systems defined by Kenyon et al. [20]. Supposing the number of vertices in the tree T grows exponentially, the explicit formula for the topological entropy of the q-MST is presented in Theorem 1.1. Next, we consider the coupled q-MST, namely, every boundary of T satisfies the q-multiple constraint and is characterized by subshift of finite type. The comparison of topological entropy for the coupled and non-coupled q-MSTs is provided in Theorem 1.4. Furthermore, the explicit formula of some coupled q-MSTs are also obtained herein. Finally, the entropy formula for the general q-MSTs, known as XΩ(S), is described in Theorem 1.6.