Functional shifts: . . .

A framework to study evolution of rules of dynamical systems is proposed with a shift-like dynamics called a functional shift. The functional shift is determined by a shift map on a set of bi-infinite sequences of some functions having the same domain and codomain. Considering the bi-infinite sequence of functions corresponding to an iterated function of a dynamical system, the functional shift allows us to analyze the dynamics of a function governing the state change of a dynamical system. Here the function is referred to as a `rule'. We study the relevance of functional shifts to sofic shifts. We prove that a class of functional shifts of nite type, having bi-infinite sequences of functions given by a shift of finite type, is equivalent to that of sofic shifts. From this theorem, we also prove that the topological entropy of general functional shifts of finite type is computable, and we provide an algorithm to compute it. Next, we show some properties of the topological entropy of rules and meta-rules. A meta-rule is a rule that governs the change of rules. Since the relation between meta-rules and rules corresponds to that of rules and states (namely, a relation of operators and operands), it is clear that states, rules, and meta-rules form a hierarchical relationship. Furthermore, since we can construct rules that govern change of meta-rules, i.e., meta-meta-rules, the hierarchical relationship continues recursively. The significant result is that the topological entropy of meta-rules provides the upper limit for that of rules. This means that the higher the level of rules in the hierarchy is, the stronger the non-linearity of these rules. The hierarchical relationship is discussed in terms of properties of rules and meta-rules. The presented framework can be used to ..