Dynamical Systems and Factors of Finite Automata

We conceive finite automata as dynamical systems on discontinuum and investigate their factors. Factors of finite automata include many well-known simple dynamical systems, e.g. hyperbolic systems and systems with finite attractors. In the quadratic family on the real interval, factors of finite automata include all systems with finite number of periodic points, as well as systems at the band-merging bifurcations. On the other hand the system with a non-chaotic attractor, which occurs at the limit of the period doubling bifurcations (at the edge of chaos), is not of this class. Next we propose as another simplicity criterion for dynamical systems the property of having chaotic limits (every point belongs to a set, whose $\omega$-limit is chaotic). We show that any factor of a finite automaton has chaotic limits but not vice versa.