Topological sequence entropy of co-induced systems

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.