Name
Abstract | ||
|---|---|---|
We study the dynamical behavior of additive D-dimensional (D >= 1) 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[38,44,41]. Among our major contributions, there is the proof that topologically transitive additive D-dimensional cellular automata over a finite abelian group are ergodic. This result represents a solid bridge between the world of measure theory and that of topology and greatly extends previous results obtained in[12,44] for linear CA over Z/mZ, i.e., additive CA in which the alphabet is the cyclic group Z/mZ and the local rules are linear combinations with coefficients in Z/mZ. 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/mZ)(n), i.e., with the local rule defined by n x n matrices with elements in Z/mZ which, in turn, strictly contains the class of linear CA over Z/mZ. In order to further emphasize that finite abelian groups are more expressive than Z/mZ we prove that, contrary to what happens in Z/mZ, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map. As a relevant consequence of our results, we have that, for additive D-dimensional CA over a finite abelian group, 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 see that invertible transitive additive CA are isomorphic to Bernoulli shifts. Furthermore, we prove that surjectivity implies openness for additive D-dimensional CA over a finite abelian group. Hence, we get that topological transitivity is equivalent to the well-known Devaney notion of chaos when D = 1. Moreover, we provide a first characterization of strong transitivity for additive CA which we suspect to be true also for the general case. (c) 2020 Elsevier B.V. All rights reserved. |
Year | DOI | Venue |
|---|---|---|
2020 | 10.1016/j.tcs.2020.06.021 | THEORETICAL COMPUTER SCIENCE |
Keywords | DocType | Volume |
Cellular automata,Additive cellular automata,Symbolic dynamics | Journal | 843 |
ISSN | Citations | PageRank |
0304-3975 | 0 | 0.34 |
References | Authors | |
0 | 4 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| alberto dennunzio | 1 | 318 | 38.17 |
| Enrico Formenti | 2 | 400 | 45.55 |
| darij grinberg | 3 | 13 | 3.65 |
| Luciano Margara | 4 | 367 | 46.16 |