The Structure of Disjoint Groups of Continuous Functions

Hojjat Farzadfard,B Khani Robati

doi:10.1155/2012/790758

Abstract

Let I be an open interval. We describe the general structure of groups of continuous self functions on I which 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 interval I. 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.

Highlights

The problem of characterizing disjoint groups of continuous functions appears in connection with the issues, such as, describing the solution of the simultaneous systems of Abel’s functional equations mainly in 1–3 and systems of differential equations with several deviations see 1, 4
One uses as a tool classes of functions φ : I → R that occur as continuous solutions of simultaneous systems of Abel equations φfxφxλf, f ∈ S, x ∈ I, 1.1 where S is a nonempty subset of Bi I the group generated by which is noncyclic and disjoint and λ : S → R is a given map
The proof we present is extracted from the proof of Theorem 3 of the Zdun’s paper 9 with a little modification

Summary

Introduction

The problem of characterizing disjoint groups of continuous functions appears in connection with the issues, such as, describing the solution of the simultaneous systems of Abel’s functional equations mainly in 1–3 and systems of differential equations with several deviations see 1, 4. One uses as a tool classes of functions φ : I → R that occur as continuous solutions of simultaneous systems of Abel equations φfxφxλf , f ∈ S, x ∈ I, 1.1 where S is a nonempty subset of Bi I the group generated by which is noncyclic and disjoint and λ : S → R is a given map. Given a Cantor-like set L fitted in I we say that φ : I → R is a Cantor function which lives on L if i φ is monotone; ii φ L R; iii φ is strictly monotone on L∗∗ Such a function φ is constant on the components of I − L∗∗, φ L∗∗ φ I − L∗∗ ∅ and φ is continuous.

Complete and Dense Disjoint Groups

The Realm of Cantor Functions

Spoiled Disjoint Groups

The Structure of Spoiled Disjoint Subgroups of Bi I

Part 2

The Structure of Spoiled Disjoint Iteration Subgroups of Bi I