On iteration groups and related functional equations
Iteration groups are considered where the group of indices is not necessarily the additive group of reals but more generally a linearly ordered Archimedean group. Thus a theorem of Hölder on these groups can be applied. A number of definitions is given and the notion of indexing the iteration group by domain is developed. A number of results is discussed in this setting. The author suggests that the literature be re- examined for possible applications of Hölder's result. One may add that the theory of iteration groups is part of the general theory of the translation equation. Over many years, Z. Moszner and his school in Cracow have developed a comprehensive theory of the translation equation, and thus of iteration groups. This extensive body of results should be taken into account when studying this problem; so should be many other results on iteration (semi-)groups. For an overview up to 1981 the reviewer's book [Topics in iteration theory (1981; Zbl 0454.39003)] could be consulted.
scientific article; zbMATH DE number 4019640
scientific article