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arXiv · 2203.09359

Borel factors and embeddings of systems in subshifts

Nishant Chandgotia·Spencer Unger
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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.

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Nishant Chandgotia, Spencer Unger. 2022-03-17. Borel factors and embeddings of systems in subshifts. https://arxiv.org/abs/2203.09359

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