Overview
This research investigates the denseness of distal points within the Bernoulli shift $2^G$ for a countable discrete group $G$. It explicitly provides an answer to a question posed by Xu and Ye, as detailed in their work "Disjointness with all minimal systems under group actions" (arxiv:2212.07830), which is slated for publication in Israel Journal of Mathematics. The study also explores a related, yet stronger, concept: almost automorphic points, and addresses a similar question regarding their denseness.
Research Context
The core of this investigation stems from a specific question by Xu and Ye concerning the denseness property of distal points in the context of Bernoulli shifts under group actions. For a countable discrete group $G$, the Bernoulli shift $2^G$ serves as the underlying space. Distal points are a class of points within dynamical systems, and their denseness is a property of interest in topological dynamics. The research also broadens its scope to include almost automorphic points, a notion that is related to distal points but is considered stronger.
Approach
The study provides characterizations related to the denseness of both distal and almost automorphic points. For almost automorphic points, it establishes that the collection of groups for which these points are dense coincides with maximal almost periodic groups. These characterizations are then leveraged to construct 2-step nilpotent groups, where the answers to the Xu-Ye question for distal points exhibit differences. In pursuit of a more intrinsic understanding of the Xu-Ye question, the research introduces a new concept: the point-distal radical for a countable discrete group. A necessary condition for the denseness property is identified as the triviality of this point-distal radical.
Further aspects of the study involve examining related questions. Specifically, it considers properties of the collection of countable groups $G$ for which the set of distal points is dense in $2^G$. It is shown that this collection is closed under finite-index extension. Additionally, the research determines that the collection of countable groups $G$ for which only constant sequences qualify as distal (or almost automorphic) points corresponds precisely to the minimally almost periodic groups.
Findings
- An answer was provided to the Xu-Ye question regarding the denseness of distal points in the Bernoulli shift $2^G$ for a countable discrete group $G$.
- For almost automorphic points, the collection of groups for which these points are dense aligns with maximal almost periodic groups.
- 2-step nilpotent groups were constructed where the answers to the Xu-Ye question for distal points diverge.
- A notion of point-distal radical for a countable discrete group was introduced.
- It was shown that a necessary condition for the denseness of distal points is for the point-distal radical to be trivial.
- The collection of countable groups $G$ for which the set of distal points is dense in $2^G$ is closed under finite-index extension.
- The collection of countable groups $G$ for which only constant sequences are distal (or almost automorphic) points coincides with minimally almost periodic groups.