Abstract
The aim of this article is to find appropriate definitions for shifts of finite type and sofic shifts in a general context of symbolic dynamics, and to study their properties. We start showing that the classical definitions of shifts of finite type and sofic shifts, as they are given in the context of finite-alphabet shift spaces on the one-dimensional monoid \(\mathbb {N}\) or \(\mathbb {Z}\) with the usual addition, do not fit for shift spaces over infinite alphabet or on other monoids. Therefore, by examining the core features in the classical definitions of shifts of finite type and sofic shifts, we propose general definitions that can be used in any context. The alternative definition given for shifts of finite type inspires the definition of a new class of shift spaces which intersects with the class of sofic shifts and includes shifts of finite type. This new class is named finitely defined shifts, and the non-finite-type shifts in it are named shifts of variable length. For the specific case of infinite-alphabet shifts on the lattice \(\mathbb {N}\) or \(\mathbb {Z}\) with the usual addition, shifts of variable length can be interpreted as the topological version of variable length Markov chains.
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Notes
Even when \(A^\mathbb {M}\) is not metrizable, it is uniformizable, that is, there exists a prodiscrete uniform structure on \(A^\mathbb {M}\)—see Section 1.9 and Appendix B in Ceccherini-Silberstein and Coornaert (2010).
In Romero et al. (2006) this class of maps was not named, although it was used the expression “generalized local rules”. In Sobottka and Gonçalves (2017) this class was designated as “generalized sliding block codes” while in Campos et al. (2021) this class of maps is referred as “extended sliding block codes” (ESBC).
A first version of this result, concerning shift spaces on the lattice \(\mathbb {Z}^d\), was given in (Romero et al. 2006, Theorem 4).
Locally finite-to-one generalized sliding block codes correspond to the “finite degree extended sliding block codes” defined in (Campos et al. 2021, Definition 2.3.ii).
In fact, in Lind and Marcus (1995) sofic shifts where defined as shift spaces generated from a directed labeled graph, which is equivalent to be the image of 1-step shifts by an 1-block SBCs (Lind and Marcus 1995, Definition 3.1.3), and then it was proved that the class of sofic shifts coincides with the class of shifts that are image of any SFT by any SBC (Lind and Marcus 1995, Theorem 3.2.1).
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Acknowledgements
This work was supported by the CNPq-Brasil grant 301445/2018-4, and developed while the author was a visiting professor at CAPES-Brasil at the Pacific Institute for the Mathematical Sciences, University of British Columbia. The author thanks Professor Brian Marcus and his research group for their hospitality. The author also thanks Charleen Stroud for her hospitality and for her help in materializing a topological example. M. Sobottka thanks the editor and the anonymous reviewer who generously donated her/his time for in-depth reading the manuscript (twice) making several valuable suggestions that improved it.
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Sobottka, M. Some Notes on the Classification of Shift Spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts. Bull Braz Math Soc, New Series 53, 981–1031 (2022). https://doi.org/10.1007/s00574-022-00292-x
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DOI: https://doi.org/10.1007/s00574-022-00292-x