On the reversibility and the closed image property of linear cellular
automata
release_r2vt7pnkhzf2bklqtfim4yddna
by
Tullio Ceccherini-Silberstein, Michel Coornaert
2010
Abstract
When G is an arbitrary group and V is a finite-dimensional vector space,
it is known that every bijective linear cellular automaton τ V^G →
V^G is reversible and that the image of every linear cellular automaton τ V^G → V^G is closed in V^G for the prodiscrete topology. In this
paper, we present a new proof of these two results which is based on the
Mittag-Leffler lemma for projective sequences of sets. We also show that if G
is a non-periodic group and V is an infinite-dimensional vector space, then
there exist a linear cellular automaton τ_1 V^G → V^G which is
bijective but not reversible and a linear cellular automaton τ_2 V^G
→ V^G whose image is not closed in V^G for the prodiscrete topology.
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