Smoothness in disjoint groups of real functions under composition
Let I be an open real interval, G a set of bijections \(I\to I\) which forms a group under composition (we will call it an iteration group). The author calls G disjoint if the graphs of two distinct elements of G do not intersect, G dense if the union of graphs is dense in \(I^ 2\) and complete if this union covers \(I^ 2\) completely. Solving a first problem of O. Borůvka and F. Neuman, the author and \textit{J. Baker} [Arch. Math., Brno 18, 121-127 (1982; Zbl 0518.26002)] proved that every dense disjoint iteration group of continuous bijections is a subgroup of a (unique) complete disjoint iteration group of continuous bijections. Their second question was whether this holds for \(C^ n\) (n\(\geq 1)\) instead of \(C^ 0.\) Here the author answers this question by constructing dense disjoint (even finitely generated) iteration groups of \(C^{\infty}\) bijections which are not subgroups of any complete disjoint complete iteration group even of \(C^ 1\) bijections. \{This paper has been submitted in 1983. Personal reasons delayed the author in proofreading. A related paper of \textit{M. A. McKiernan} appeared in Aequationes Math. 31, 37-46 (1986; Zbl 0627.39004).\}
dense disjoint iteration group of continuous bijections
scientific article; zbMATH DE number 4150555