On the Expressive Power of Quasiperiodic SFT

Abstract

In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in Z^d.  The minimal shifts are those shifts in which all configurations contain exactly  the same patterns. Two  classes of  shifts play a prominent role in symbolic dynamics, in language theory  and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns). 
We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by 1) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E. Jeandel.  We  prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which  Kolmogorov complexity  of  all patterns of size n\times n is \Omega(n).