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


26C15 Real rational functions
68Q45 Formal languages and automata