Regular unimodal systems and factors of finite automata

Abstract

Dynamical systems at the edge of chaos, which have been considered as models of self-organization phenomena, are marked by their ability to perform nontrivial computations. To distinguish them from systems with limited computing power, we formulate two simplicity criteria for general dynamical systems, and apply them to unimodal systems on real interval. We say that a dynamical system is regular, if it yields a regular language when observed through arbitrary almost disjoint cover. Finite automata are regarded as dynamical systems on zero-dimensional spaces and their factors yield another class of simple dynamical systems. These two criteria coincide on subshifts, since a subshift is regular iff it is a factor of a finite automaton (sofic systems). A unimodal system on real interval is regular if it has only a finite number of periodic points, and nonrecursive otherwise. On the other hand each $S$-unimodal system with finite, periodic or preperiodic kneading sequence is a factor of a finite automaton. Thus preperiodic $S$-unimodal systems are factors of finite automata, which are not regular.