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One dimensional dynamics and factors of finite automata. (English) Zbl 0806.58016

The author considers dynamical systems \((X,f)\), where \(X\) is a compact metric space and \(f: X \to X\) a continuous map. If \((X_ 1,f_ 1)\) and \((X_ 2,f_ 2)\) are two dynamical systems one calls a continuous map \(\varphi : X_ 1 \to X_ 2\) a homomorphism if \(f_ 2 \varphi = \varphi f_ 1\). Finally one says the system \((X_ 2,f_ 2)\) is a factor of \((X_ 1,f_ 1)\) if \(\varphi : X_ 1 \to X_ 2\) is a surjective homomorphism. Every dynamical system on a compact metric space is a factor of some dynamical system on discontinuum. Special dynamical systems on discontinuum are the so-called finite automata.
The author shows that a large class of dynamical systems are factors of finite automata. In particular it is shown that any homeomorphism of the real interval is of this class. Moreover it is shown that an orientation preserving homeomorphism of the circle is a factor of a finite automaton iff its rotation number is rational and any \(S\)-unimodal system on a real interval with preperiodic or odd periodic kneading sequence is a factor of a finite automaton.

MSC:

37E99 Low-dimensional dynamical systems
37B99 Topological dynamics
68Q70 Algebraic theory of languages and automata
18B20 Categories of machines, automata
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