Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties

We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e.with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map. As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case

From Linear to Additive Cellular Automata

This paper proves the decidability of several important properties of additive cellular automata over finite abelian groups. First of all, we prove that equicontinuity and sensitivity to initial conditions are decidable for a nontrivial subclass of additive cellular automata, namely, the linear cellular automata over \u207f, where is the ring \u2124/m\u2124. The proof of this last result has required to prove a general result on the powers of matrices over a commutative ring which is of interest in its own. Then, we extend the decidability result concerning sensitivity and equicontinuity to the whole class of additive cellular automata over a finite abelian group and for such a class we also prove the decidability of topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to it. Finally, a decidable characterization of injectivity and surjectivity for additive cellular automata over a finite abelian group is provided in terms of injectivity and surjectivity of an associated linear cellular automata over \u207f

On Computing the Entropy of Cellular Automata

We study the topological entropy of a particular class of dynamical systems: cellular automata. The topological entropy of a dynamical system (X,F) is a measure of the complexity of the dynamics of F over the space X. The problem of computing (or even approximating) the topological entropy of a given cellular automata is algorithmically undecidable (Ergodic Theory Dynamical Systems 12 (1992) 255). In this paper, we show how to compute the entropy of two important classes of cellular automata namely, linear and positively expansive cellular automata. In particular, we prove a closed formula for the topological entropy of D-dimensional (D?1) linear cellular automata over the ring and we provide an algorithm for computing the topological entropy of positively expansive cellular automata

Subshifts with Simple Cellular Automata

A subshift is a set of infinite one- or two-way sequences over a fixed finite set, defined by a set of forbidden patterns. In this thesis, we study subshifts in the topological setting, where the natural morphisms between them are ones defined by a (spatially uniform) local rule. Endomorphisms of subshifts are called cellular automata, and we call the set of cellular automata on a subshift its endomorphism monoid. It is known that the set of all sequences (the full shift) allows cellular automata with complex dynamical and computational properties. We are interested in subshifts that do not support such cellular automata. In particular, we study countable subshifts, minimal subshifts and subshifts with additional universal algebraic structure that cellular automata need to respect, and investigate certain criteria of ‘simplicity’ of the endomorphism monoid, for each of them. In the case of countable subshifts, we concentrate on countable sofic shifts, that is, countable subshifts defined by a finite state automaton. We develop some general tools for studying cellular automata on such subshifts, and show that nilpotency and periodicity of cellular automata are decidable properties, and positive expansivity is impossible. Nevertheless, we also prove various undecidability results, by simulating counter machines with cellular automata. We prove that minimal subshifts generated by primitive Pisot substitutions only support virtually cyclic automorphism groups, and give an example of a Toeplitz subshift whose automorphism group is not finitely generated. In the algebraic setting, we study the centralizers of CA, and group and lattice homomorphic CA. In particular, we obtain results about centralizers of symbol permutations and bipermutive CA, and their connections with group structures.Siirretty Doriast