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On the expressive power of quasiperiodic SFT

Andrei Romashchenko 1 Bruno Durand 1
1 ESCAPE - Systèmes complexes, automates et pavages
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in $\mathbb{Z}^d$. 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 $\Omega(n)$.
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Submitted on : Wednesday, October 25, 2017 - 10:41:24 AM
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Andrei Romashchenko, Bruno Durand. On the expressive power of quasiperiodic SFT. MFCS: Mathematical Foundations of Computer Science, Aug 2017, Aalborg, Denmark. pp.5:1-5:14, ⟨10.4230/LIPIcs.MFCS.2017.5⟩. ⟨lirmm-01623207⟩

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