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A Garden of Eden theorem for linear subshifts

Published online by Cambridge University Press:  13 June 2011

TULLIO CECCHERINI-SILBERSTEIN  and
MICHEL COORNAERT
TULLIO CECCHERINI-SILBERSTEIN
Affiliation:
Dipartimento di Ingegneria, Università del Sannio, Corso Garibaldi 107, 82100 Benevento, Italy (email: [email protected])
MICHEL COORNAERT
Affiliation:
Institut de Recherche Mathématique Avancée, Université de Strasbourg et CRNS, 7 rue René-Descartes, 67000 Strasbourg, France (email: [email protected])
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Abstract

Let G be an amenable group and let V be a finite-dimensional vector space over an arbitrary field 𝕂. We prove that if X⊂VG is a strongly irreducible linear subshift of finite type and τ:X→X is a linear cellular automaton, then τ is surjective if and only if it is pre-injective. We also prove that if G is countable and X⊂VG is a strongly irreducible linear subshift, then every injective linear cellular automaton τ:X→X is surjective.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 81 - 102
Copyright
Copyright © Cambridge University Press 2011

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