Strong marker sets and applications

TL;DR

The paper proves the existence of special clopen marker sets with regularity properties in certain dynamical systems, leading to applications like clopen tree sections and edge colorings, with implications for generating sets in dimension 2.

Contribution

It introduces strong marker sets with regularity properties in $F(2^{bZ^n})$, providing new proofs for edge colorings and extending results to general generating sets.

Findings

Existence of clopen marker sets with regularity properties.

Construction of clopen tree sections in $F(2^{bZ^n})$.

Existence of continuous edge colorings for Schreier graphs.

Abstract

We prove the existence of clopen marker sets with some strong regularity property. For each $n\geq 1$ and any integer $d\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\mathbb{Z}^n})$ such that (1) for any distinct $x,y\in M$ in the same orbit, $\rho(x,y)\geq d$; (2) for any $1\leq i\leq n$ and any $x\in F(2^{\mathbb{Z}^n})$, there are non-negative integers $a, b\leq D$ such that $a\cdot x\in M$ and $-b\cdot x\in M$. As an application, we obtain a clopen tree section for $F(2^{\mathbb{Z}^n})$. Based on the strong marker sets, we get a quick proof that there exist clopen continuous edge $(2n+1)$-colorings of $F(2^{\mathbb{Z}^n})$. We also consider a similar strong markers theorem for more general generating sets. In dimension 2, this gives another proof of the fact that for any generating set $S\subseteq \mathbb{Z}^2$, there is a continuous proper edge $(2|S|+1)$-coloring of the Schreier graph of $F(2^{\mathbb{Z}^n})$ with generating set $S$.