Factor maps and embeddings for random ${\mathbb{Z}}^d$ shifts of finite type

Abstract

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

References in corpus (1)

• Factoring onto ${\mathbb{Z}}^d$ subshifts with the finite extension property