The Structure of Disjoint Groups of Continuous Functions

LetIbe an open interval. We describe the general structure of groups of continuous self functions onIwhich are disjoint, that is, the graphs of any two distinct elements of them do not intersect. Initially the class of all disjoint groups of continuous functions is divided in three subclasses: cyclic groups, groups the limit points of their orbits are Cantor-like sets, and finally those the limit points of their orbits are the whole intervalI. We will show that (1) each group of the second type is conjugate, via a specific homeomorphism, to a piecewise linear group of the same type; (2) each group of the third type is a subgroup of a continuous disjoint iteration group. We conclude the Zdun's result on the structure of disjoint iteration groups of continuous functions as special case of our results.

On Covariant Embeddings of a Linear Functional Equation with Respect to an Analytic Iteration Group

Let a(x), b(x), p(x) be formal power series in the indeterminate x over [Formula: see text] (i.e. elements of the ring [Formula: see text] of such series) such that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iteration group [Formula: see text] in [Formula: see text]. By a covariant embedding of the linear functional equation [Formula: see text] (for the unknown series [Formula: see text]) with respect to [Formula: see text]. In this paper we solve the system ((Co1), (Co2)) (of so-called cocycle equations) completely, describe when and how the boundary conditions (B1) and (B2) can be satisfied, and present a large class of equations (L) together with iteration groups [Formula: see text] for which there exist covariant embeddings of (L) with respect to [Formula: see text].