Abstract. For d ≥ 2 we exhibit mixing 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.

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Since seminal work of Bowen ([2]), it has been known that the specification property implies various useful properties about an expansive topological dynamical system, among them uniqueness of the measure of maximal entropy (often referred to as intrinsic ergodicity). Weakenings of the specification property have been defined and profitably applied in various works such as [6], [9], [11], [16], and [17]. It has been an open question (see p. 798 of [4]) whether two of these properties, which we here call almost specification and non-uniform specification, imply intrinsic ergodicity for expansive topological systems. We answer this question negatively by exhibiting examples of subshifts with multiple measures of maximal entropy with disjoint support which have non-uniform specification with any gap function f (n) = O(ln n) or almost specification with any mistake function g(n) ≥ 4. We also show some results in the opposite direction, showing that subshifts with non-uniform specification with gap function f (n) = o(ln n) or almost specification with mistake function g(n) = 1 cannot have multiple measures of maximal entropy with disjoint support.

Motivated by Hochman's notion of subdynamics of a Z d subshift (2009), we define and examine the projective subdynamics of Z d shifts of finite type (SFTs) where we restrict not only the action but also the phase space. We show that any Z sofic shift of positive entropy is the projective subdynamics of a Z 2 (Z d) SFT, and that there is a simple condition characterizing the class of zero-entropy Z sofic shifts which are not the projective subdynamics of any Z 2 SFT. We define notions of stable and unstable subdynamics in analogy with the notions of stable and unstable limit sets in cellular automata theory, and discuss how our results fit into this framework. One-dimensional strictly sofic shifts of positive entropy admit both a stable and an unstable realization, whereas Z SFTs only allow for stable realizations and a particular class of zero-entropy proper Z sofics only allows for an unstable realization. Finally, we prove that the union of finitely many Z k subshifts, all of which are realizable in Z d SFTs, is again realizable when it contains at least two periodic points, that the projective subdynamics of Z 2 SFTs with the uniform filling property (UFP) are always stable, thus sofic, and we exhibit a class of non-sofic Z subshifts which are not the projective subdynamics of any Z d SFT.

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