[Paper Review] Topological entropy of continuous actions of compactly generated groups

Friedrich Martin Schneider|arXiv (Cornell University)|Feb 13, 2015

Mathematical Dynamics and Fractals13 references3 citations

TL;DR

This paper introduces a new notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It proves that vanishing topological entropy implies amenability and establishes that the canonical action of a compactly generated locally compact group $G$ on the weak-* unit ball of $L^∞(G)$ has vanishing entropy if and only if $G$ is compact, and infinite entropy if $G$ is almost connected and non-compact.

ABSTRACT

We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It is shown that any continuous action of a compactly generated topological group on a compact Hausdorff space with vanishing topological entropy is amenable. Given an arbitrary compactly generated locally compact Hausdorff topological group $G$, we consider the canonical action of $G$ on the closed unit ball of $L^{1}(G)' \cong L^{\infty}(G)$ endowed with the corresponding weak-$^{\ast}$ topology. We prove that this action has vanishing topological entropy if and only if $G$ is compact. Furthermore, we show that the considered action has infinite topological entropy if $G$ is almost connected and non-compact.

Motivation & Objective

• To extend the concept of topological entropy to continuous actions of compactly generated topological groups on compact Hausdorff spaces.
• To establish a connection between vanishing topological entropy and amenability of group actions.
• To characterize the topological entropy of the canonical action of a compactly generated locally compact group $G$ on the weak-* unit ball of $L^\infty(G)$.
• To determine when this canonical action has vanishing or infinite topological entropy based on the group's structure.

Proposed method

• Introduces a new definition of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces using finite open covers and their refinements.
• Applies the concept of $S^n$-refinement of open covers under group actions to measure the growth of covering complexity.
• Uses the dual space $L^1(G)' \cong L^\infty(G)$ and equips the closed unit ball with the weak-* topology to define a canonical action of $G$.
• Employs compact generating sets $S$ and analyzes the growth rate of covering numbers $(S^n:\mathcal{V})_\alpha$ to define the entropy quantity $\eta(\alpha,S,\mathcal{V})$.
• Applies Lemma 2.2 to ensure that compact sets are contained in the interior of higher powers of a generating set.
• Uses the structure of almost connected groups and the non-compactness of $\bigcup_n U^n$ for identity neighborhoods $U$ to construct covers with exponential growth in refinement complexity.

Experimental results

Research questions

• RQ1 Does vanishing topological entropy imply amenability for continuous actions of compactly generated topological groups on compact Hausdorff spaces?
• RQ2 For the canonical action of a compactly generated locally compact group $G$ on the weak-* unit ball of $L^\infty(G)$, when does the topological entropy vanish?
• RQ3 When does this canonical action have infinite topological entropy?
• RQ4 How does the topological entropy of the canonical action relate to the compactness of $G$?
• RQ5 What is the relationship between the almost connectedness of $G$ and the entropy of its canonical action?

Key findings

• Vanishing topological entropy of a continuous action of a compactly generated topological group on a compact Hausdorff space implies that the action is amenable.
• The canonical action $\alpha$ of a compactly generated locally compact Hausdorff group $G$ on the weak-* unit ball of $L^\infty(G)$ has vanishing topological entropy if and only if $G$ is compact.
• If $G$ is almost connected and non-compact, then the canonical action $\alpha$ has infinite topological entropy.
• For any non-compact $G$, there exists a two-element open covering $\mathcal{V}$ such that $\eta(\alpha,S,\mathcal{V}) \geq \frac{1}{2k}$, where $k$ is the minimal $n$ such that $S^n$ is an identity neighborhood.
• The entropy quantity $\eta(\alpha,S,\mathcal{V})$ can be made arbitrarily large for almost connected non-compact $G$, proving infinite topological entropy.