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. Also, we will discuss an example where $H$ is a proper subgroup of $G$ with finite index where the maximal pattern entropy of the $H$ action is equal to the co-induced action of $G$. 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.

Figures (1)

• Figure 3.1: $(A,\mathbb{Z},T_1)$

Theorems & Definitions (42)

• Proposition 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.14
• Theorem 2.1: Huang and Ye, Theorem A.1 Huang_Ye_maximal_sequence_entropy
• Theorem 2.2: Huang and Ye, Theorem 2.3(2) Huang_Ye_maximal_sequence_entropy
• Proposition 2.3: Huang and Ye, Lemma 2.5 Huang_Ye_maximal_sequence_entropy
• Theorem 2.4: Kerr and Li, Proposition 5.4 Kerr_Li_Independence
• Theorem 2.5: Huang, Li, Shao, Ye, Theorem 2.1 Huang_Shao_Ye_sequence_entropy_pairs