Factor maps and embeddings for random ℤd shifts of finite type

Abstract

For any d ≥ 1, random ℤd shifts of finite type (SFTs) were defined in previous work of the authors. For a parameter α ∈ [0, 1], an alphabet $$\mathcal{A}$$ , and a scale n ∈ ℕ, one obtains a distribution of random ℤd SFTs by randomly and independently forbidding each pattern of shape {1,..., n}d with probability 1 − α from the full shift on $$\mathcal{A}$$ . We prove twomain results concerning random ℤd SFTs. First, we establish sufficient conditions on α, $$\mathcal{A}$$ , and a ℤd subshift Y so that a random ℤd SFT factors onto Y with probability tending to one as n tends to infinity. Second, we provide sufficient conditions on α, $$\mathcal{A}$$ and a ℤd subshift X so that X embeds into a random ℤd SFT with probability tending to one as n tends to infinity.