Multidimensional sofic shifts without separation and their factors

Abstract

For d ≥ 2 d\geq 2 we exhibit mixing Z d \mathbb {Z}^d shifts of finite type and sofic shifts with large entropy but poorly separated subsystems (in the sofic examples, the only minimal subsystem is a single point). These examples consequently have very constrained factors; in particular, no non-trivial full shift is a factor. We also provide examples to distinguish certain mixing conditions and develop the natural class of “block gluing” shifts. In particular, we show that block gluing shifts factor onto all full shifts of strictly smaller entropy.