Borel Combinatorics of Schreier Graphs of $\mathbb{Z}$-actions

In this paper we consider the Borel combinatorics of Schreier graphs of $\mathbb{Z}$-actions with arbitrary finite generating sets. We formulate the Borel combinatorics in terms of existence of Borel equivariant maps from $F(2^{\mathbb{Z}})$ to subshifts of finite type. We then show that the Borel combinatorics and the continuous combinatorics coincide, and both are decidable. This is in contrast with the case of $\mathbb{Z}^2$-actions. We then turn to the problem of computing Borel chromatic numbers for such graphs. We give an algorithm for this problem which runs in exponential time. We then prove some bounds for the Borel chromatic numbers and give a formula for the case where the generating set has size 4.