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On fixed points of rational functions with values on a circle. (English) Zbl 0706.26012
The author proves the following theorem: Let U be a nonempty open subset of the real Euclidean plane and $$C\subset U$$ a circular line. Let f: $$U\to C$$ be a map such that $f(u)=(g_ 1(u)/h_ 1(u),g_ 2(u)/h_ 2(u)),\quad h_ 1(u)\neq 0,\quad h_ 2(u)\neq 0\text{ for all } u\in U,$ $$g_ 1$$, $$g_ 2$$, $$h_ 1$$, $$h_ 2$$ being polynomials. Then f has a fixed point $$\tilde u\in C.$$
After proving this theorem the author applies it to an automata theoretical theorem; the underlying abstract automata simulate the behaviour of a human drawer (constructions with compass and ruler and constructions with rectangular ruler).
In the paper are several unpleasant misprints.
Reviewer: M.Jůza

##### MSC:
 26C15 Real rational functions 68Q45 Formal languages and automata