Factor maps and embeddings for random ℤ^d shifts of finite type

Abstract

For any d ≥ 1, random ℤ^d 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 ℤ^d 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 ℤ^d SFTs. First, we establish sufficient conditions on α, $\mathcal{A}$, and a ℤ^d subshift Y so that a random ℤ^d 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 ℤ^d subshift X so that X embeds into a random ℤ^d SFT with probability tending to one as n tends to infinity.