Dynamical quasitilings of amenable group

TL;DR

This paper constructs a dynamical quasitiling for free actions of infinite countable amenable groups on zero-dimensional compact spaces, with properties ensuring high coverage, continuity, and F{4}lner conditions, and establishes a factor map structure.

Contribution

It introduces a method to assign quasitilings with desirable dynamical and combinatorial properties to points in zero-dimensional spaces under amenable group actions.

Findings

Existence of quasitilings with high lower Banach Density

Quasitilings have shapes large relative to F{4}lner sets

The assignment defines a factor map from the space to a shift space

Abstract

We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, F{\o}lner and dynamical properties, i.e to every $x\in X$ we can assign a quasitiling $\mathcal{T}_x$ of $G$ (with all the $\mathcal{T}_x$ using the same, finite set of shapes) such that the tiles of $\mathcal{T}_x$ are disjoint, their union has arbitrarily high lower Banach Density, all the shapes of $\mathcal{T}_x$ are large subsets of an arbitrarily large F{\o}lner set, and if we consider $\mathcal{T}_x$ to be an element of a shift space over a certain finite alphabet, then the mapping $x\mapsto \mathcal{T}_x$ is a factor map.