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Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Part of: Theory of computing Topological dynamics Algebraic geometry: Foundations

Published online by Cambridge University Press:  15 February 2024

TULLIO CECCHERINI-SILBERSTEIN Open the ORCID record for TULLIO CECCHERINI-SILBERSTEIN [Opens in a new window] ,
MICHEL COORNAERT  and
XUAN KIEN PHUNG Open the ORCID record for XUAN KIEN PHUNG [Opens in a new window]
TULLIO CECCHERINI-SILBERSTEIN*
Affiliation:
Dipartimento di Ingegneria, Università del Sannio, 82100 Benevento, Italy
MICHEL COORNAERT
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, F-67000 Strasbourg, France (e-mail: michel.coornaert@math.unistra.fr)
XUAN KIEN PHUNG
Affiliation:
Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, Montréal, Québec H3T 1J4, Canada Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, H3T 1J4, Canada (e-mail: phungxuankien1@gmail.com)
*
e-mail: tullio.cs@sbai.uniroma1.it
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Abstract

Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $. We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

Keywords

algebraic varietyalgebraic cellular automatonalgebraic sofic subshiftnilpotencylimit set

MSC classification

Primary: 37B15: Cellular automata 14A10: Varieties and morphisms
Secondary: 37B10: Symbolic dynamics 37B20: Notions of recurrence 14A15: Schemes and morphisms 68Q80: Cellular automata
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 42
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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