One-dimensional dynamics and factors of finite automata

We argue that simple dynamical systems are factors of finite automata, regarded as dynamical systems on discontinuum. We show that any homeomorphism of the real interval is of this class. An orientation preserving homeomorphism of the circle is a factor of a finite automaton iff its rotation number is rational. Any $S$-unimodal system on the real interval, whose kneading sequence is either periodic odd or preperiodic, is also a factor of a finite automaton, while $S$-unimodal systems at limits of period doubling bifurcations are not.