  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.
