Topological sequence entropy of co-induced systems

Abstract

Let $G$ be a discrete, countably infinite group and $H$ a subgroup of $G$. If $H$ acts continuously on a compact metric space $X$, then we can induce a continuous action of $G$ on ${\prod }_{H\backslash G}X$, where $H\backslash G$ is the collection of right cosets of $H$ in $G$. This process is known as the co-induction. In this article, we will calculate the maximal pattern entropy of the co-induction. If $[G:H]<+\infty$, we will show that the $H$ action is null if and only if the co-induced action of $G$ is null. Additionally, we will discuss an example where $H$ is a proper subgroup of $G$ with finite index, and we will show that the maximal pattern entropy of the $H$-action on $X$ is equal to the maximal pattern entropy of the co-induced action of $G$ on ${\prod }_{H\backslash G}X$. If $[G:H]=+\infty$, we will show that the maximal pattern entropy of the co-induction is always $+\infty$ given the $H$-system is not trivial.