Borel factors and embeddings of systems in subshifts

TL;DR

This paper explores the combinatorics of free Borel actions of  groups on Polish spaces, introducing property F to facilitate embeddings of subshifts, with applications to coloring and tiling problems.

Contribution

It introduces property F for  shift spaces, enabling Borel embeddings of subshifts under certain conditions, extending previous results and answering open questions.

Findings

Property F holds for several natural examples including colorings and tilings.

Any subshift (modulo periodic points) can be Borel embedded into spaces with property F under entropic conditions.

The work generalizes and recovers several known results in the field.

Abstract

In this paper we study the combinatorics of free Borel actions of the group $\mathbb Z^d$ on Polish spaces. Building upon recent work by Chandgotia and Meyerovitch, we introduce property $F$ on $\mathbb Z^d$-shift spaces $X$ under which there is an equivariant map from any free Borel action to the free part of $X$. Under further entropic assumptions, we prove that any subshift $Y$ (modulo the periodic points) can be Borel embedded into $X$. Several examples satisfy property $F$ including, but not limited to, the space of proper $3$-colourings, tilings by rectangles (under a natural arithmetic condition), proper $2d$-edge colourings of $\mathbb Z^d$ and the space of bi-infinite Hamiltonian paths. This answers questions raised by Seward, and Gao-Jackson, and recovers a result by Weilacher and some results announced by Gao-Jackson-Krohne-Seward.