Turing Degree Spectra of Minimal Subshifts

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for the effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension d can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension d + 1. Supported by the ANR grant Racaf ANR-15-CE40-0016-01. Preliminary versions of some of the presented results we published in conference papers on MFCS-2015 [23] (Theorems 3-4) and MFCS-2017 [26] (Theorems 6-7).

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“…Remark 1. The Turing degree spectrum of a non effective minimal shift can be much more complex than what we get in Theorem 4, see [27].…”

Section: Quasiperiodicity Is Compatible With Non-computabilitymentioning

confidence: 88%

The expressiveness of quasiperiodic and minimal shifts of finite type

Durand

Romashchenko

2020

Ergod. Th. Dynam. Sys.

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“…One of the results, however, applied to minimal subshifts in any dimension: namely, such a subshift's spectrum either consists only of 0 (the degree of computable sequences) or must contain the cone above any of its degrees (this result is recalled in Lemma 2.3). It was later proved that spectra of minimal subshifts may contain several cones [5] and then that they actually correspond exactly to the enumeration cones of co-total enumeration degrees [9].…”

Section: Ronnie Pavlov and Pascal Vaniermentioning

confidence: 99%

The relationship between word complexity and computational complexity in subshifts

Pavlov

Vanier²

2021

Discrete &Amp; Continuous Dynamical Systems - A

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We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree 0 or is a union of cones is realizable by subshifts with a wide range of 'intermediate' complexity growth rates between linear and exponential.

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“…One of the results, however, applied to minimal subshifts in any dimension: namely, such a subshift's spectrum either consists only of 0 (the degree of computable sequences) or must contain the cone above any of its degrees (this result is recalled in Lemma 2.4). It was later proved that spectra of minimal subshifts may contain several cones [HV17] and then that they actually correspond exactly to the enumeration cones of co-total enumeration degrees [McC18].…”

Section: Introductionmentioning

confidence: 99%