Linear Cellular Automata

Abstract

In this chapter we study linear cellular automata, namely cellular automata whose alphabet is a vector space and which are linear with respect to the induced vector space structure on the set of configurations. If the alphabet vector space and the underlying group are fixed, the set of linear cellular automata is a subalgebra of the endomorphism algebra of the configuration space (Proposition 8.1.4). An important property of linear cellular automata is that the image of a finitely supported configuration by a linear cellular automaton also has finite support (Proposition 8.2.3). Moreover, a linear cellular automaton is entirely determined by its restriction to the space of finitely-supported configurations (Proposition 8.2.4) and it is pre-injective if and only if this restriction is injective (Proposition 8.2.5). The algebra of linear cellular automata is naturally isomorphic to the group algebra of the underlying group with coefficients in the endomorphism algebra of the alphabet vector space (Theorem 8.5.2). Linear cellular automata may be also regarded as endomorphisms of the space of finitely-supported configurations, viewed as a module over the group algebra of the underlying group with coefficients in the ground field (Proposition 8.7.5). This representation of linear cellular automata is always one-to-one and, when the alphabet vector space is finite-dimensional, it is also onto (Theorem 8.7.6). The image of a linear cellular automaton is closed in the space of configurations for the prodiscrete topology, provided that the alphabet is finite dimensional (Theorem 8.8.1). We exhibit an example showing that if one drops the finite dimensionality of the alphabet, then the image of a linear cellular automaton may fail to be closed. In Sect. 8.9 we prove a linear version of the Garden of Eden theorem. For the proof, we introduce the mean dimension of a vector subspace of the configuration space. We show that for a linear cellular automaton with finite-dimensional alphabet, both pre-injectivity and surjectivity are equivalent to the maximality of the mean dimension of the image of the cellular automaton (Theorem 8.9.6). We exhibit two examples of linear cellular automata with finite-dimensional alphabet over the free group of rank two, one which is pre-injective but not surjective, and one which is surjective but not pre-injective. This shows that the linear version of the Garden of Eden theorem fails to hold for groups containing nonabelian free subgroups (see Sects. 8.10 and 8.11). Provided the alphabet is finite dimensional, the inverse of every bijective linear cellular automaton is also a linear cellular automaton (Corollary 8.12.2). In Sect. 8.13 we study the pre-injectivity and surjectivity of the discrete Laplacian over the real numbers and prove a Garden of Eden type theorem (Theorem 8.13.2) for such linear cellular automata with no amenability assumptions on the underlying group. As an application, we deduce a characterization of locally finite groups in terms of real linear cellular automata (Corollary 8.13.4). In Sect. 8.14 we define linear surjunctivity and prove that all sofic groups are linearly surjunctive (Theorem 8.14.4). The notion of stable finiteness for rings is introduced in Sect. 8.15. A stably finite ring is a ring for which one-sided invertible square matrices are also two-sided invertible. It is shown that linear surjunctivity is equivalent to stable finiteness of the associated group algebra (Corollary 8.15.6). As a consequence, we deduce that group algebras of sofic groups are stably finite for any ground field (Corollary 8.15.8). In the last section, we prove that the absence of zero-divisors in the group algebra of an arbitrary group is equivalent to the fact that every non-identically-zero linear cellular automaton with one-dimensional alphabet is pre-injective (Corollary 8.16.12).

We recall that in this book all rings are assumed to be associative (but not necessarily commutative) with a unity element, and that a field is a nonzero commutative ring in which each nonzero element is invertible.