Ergodicity of linear cellular automata over $\text{Z}$_m

We continue the study of cellular automata (CA) directional dynamics, i.e. , the behavior of the joint action of CA and shift maps. This notion has been investigated for general CA in the case of expansive dynamics by Boyle and Lind; and by Sablik for sensitivity and equicontinuity. In this paper we give a detailed classification for the class of additive CA providing non-trivial examples for some classes of Sablik’s classification. Moreover, we extend the directional dynamics studies by considering also factor languages and attractors.

In this paper, we investigate some ergodic properties of Z²-actions T_p,n generated by an additive cellular automata and shift acting on the space of all doubly-infinitive sequences taking values in Z_m.

We study two dynamical properties of linear D-dimensional cellular automata over $\text{Z}{}_{\mathrm{m}}$ namely, denseness of periodic points and topological mixing. For what concerns denseness of periodic points, we complete the work initiated in (Theoret. Comput. Sci. 174 (1997) 157, Theoret. Comput. Sci. 233 (1–2) (2000) 147, 14th Annual Symp. on Theoretical Aspects of Computer Science (STACS ’97), LNCS n. 1200, Springer, Berlin, 1997, pp. 427–438) by proving that a linear cellular automata has dense periodic points over the entire space of configurations if and only if it is surjective (as conjectured in (Cattaneo et al., 2000)). For non-surjective linear CA we give a complete characterization of the subspace where periodic points are dense. For what concerns topological mixing, we prove that this property is equivalent to transitivity and then easily checkable. Finally, we classify linear cellular automata according to the definition of chaos given by Devaney in (An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, Reading, MA, USA, 1989).

We study the dynamical behavior of $\text{D}$-dimensional linear cellular automata over $\text{Z}{}_{\mathrm{m}}$. We provide an easy-to-check necessary and sufficient condition for a $\text{D}$-dimensional linear cellular automata over $\text{Z}{}_{\mathrm{m}}$ to be ergodic and topologically transitive. As a byproduct, we get that for linear cellular automata ergodicity is equivalent to topological transitivity. Finally, we prove that for 1-dimensional linear cellular automata over $\text{Z}{}_{\mathrm{m}}$, regularity (denseness of periodic orbits) is equivalent to surjectivity.

We study the dynamical behavior of D-dimensional linear cellular automata over Z_m. We provide easy-to-check necessary and sufficient conditions for a D-dimensional linear cellular automata over Z_m to be sensitive to initial conditions, positively expansive, strongly transitive, and equicontinuous. As a consequence of our results, we have a complete and efficiently computable topological classification of D-dimensional linear cellular automata over Z_m according to the most important dynamical properties studied in the theory of discrete time dynamical systems.

In this paper we study the asymptotic behavior of D-dimensional linear cellular automata over the ring Z_m (D⩾1, m⩾2). In the first part of the paper we consider nonsurjective cellular automata (CA). We prove that, after a transient phase of length at most ⌊log₂ m⌋, the evolution of a linear nonsurjective cellular automata F takes place completely within a subspace Y_F. This result suggests that we can get valuable information on the long term behavior of F by studying its properties when restricted to Y_F. We prove that such study is possible by showing that the system (Y_F, F) is topologically conjugated to a linear cellular automata F* defined over a different ring Z_m. In the second part of the paper, we study the attractor sets of linear cellular automata. Recently, Kurka has shown that CA can be partitioned into five disjoint classes according to the structure of their attractors. We present a procedure for deciding the membership in Kurka's classes for any linear cellular automata. Our procedure requires only gcd computations involving the coefficients of the local rule associated to the cellular automata.