Non-uniform cellular automata and distributions of rules

Cellular automata (CA) are discrete dynamical systems consisting in an infinite number of finite automata arranged on a regular lattice. Each node of the lattice contains a variable which can only take a finite number of different values. At any time, the state of the CA, also called its current configuration, is specified by the values of those variables. A CA make evolve its configuration in discrete time by applying a local rule simultaneously to all the variables in the lattice. The value of a variable in a new configuration is computed from the values of a finite number of variables of its neighborhood in the previous configuration according to the local rule. CA have been studied and used in a number of disciplines (computer science, mathematics, physic, biology, chemistry) with different purposes (representing natural phenomena, pseudo-random number generation, cryptography). However the fact that a same rule is applied everywhere in the lattice can be a constraint in some cases (singular behavior, boundary conditions, non-resiliency). Then variants of CA have been introduced to allow the local rule used to compute the value of a variable to depend of its position. Those variants are called non-uniform cellular automata (ν-CA) or hybrid cellular automata (HCA) [1]. In this paper we study ν-CA on one-dimensional lattice defined over a finite set of local rules. The main goal is to determine how the local rules can be mixed to ensure the produced ν-CA has some properties. In a first part, we give some background for the study of ν-CA. Then surjectivity and injectivity are studied using a variant of DeBruijn graphs. The next part is dedicated to the number-conserving property. In a last section, we will be interested in the dynamical properties for linear ν-CA.