Classification of sofic projective subdynamics of multidimensional shifts of finite type

Ronnie Pavlov, Michael Schraudner
2014 Transactions of the American Mathematical Society  
Motivated by Hochman's notion of subdynamics of a Z d subshift [8] , 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
more » ... d 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 a particular class of zero-entropy Z sofics only allows for an unstable realization. Finally, we prove that the union of 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 sofic and we exhibit a class of non-sofic Z subshifts which are not the subdynamics of any Z d SFT.
doi:10.1090/s0002-9947-2014-06259-4 fatcat:wrowsd4rpzdevac66gfolaabqm