Borel factors and embeddings of systems in subshifts

Abstract

In this paper we study the combinatorics of free Borel actions of the group Zd on Polish spaces. Building upon recent work by Chandgotia and Meyerovitch, we introduce property F on Zd-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 Zd 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.