Simplicity criteria for dynamical systems

Abstract

We formulate two simplicity criteria for dynamical systems based on the concepts of finite automata and regular languages. Finite automata are regarded as dynamical systems on discontinuum and their factors yield the first simplicity class. A finite cover of a topological space is almost disjoint, if it consists of closed sets which have the same dimension as the space, and meet in sets whose dimension is smaller. A dynamical system is regular, if it yields a regular language when observed through any finite almost disjoint cover. Next we formulate two topological simplicity criteria. A dynamical system has finite attractors, if the Ω-limit of its state space is finite. A dynamical system has chaotic limits, if every point is included in a set whose Ω-limit is either a finite orbit or a chaotic subsystem. We show the relations between these criteria and classify according to them several classes of zero-dimensional and one-dimensional dynamical systems.