Let

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 [

Throughout this paper

The following two theorems, which have been proved in [

Let

for all

for all

the set

Let

By virtue of these theorems one divides disjoint subgroups of

One uses as a tool classes of functions

Given a Cantor-like set

Such a function

Given a function

Let

Theorem 12 of [

Let

In Section

Theorems

If

Conversely, to every complete disjoint subgroup

For the first part of the theorem (whose proof is straightforward) see [

Our next goal is to present a description of dense disjoint subgroups of

Let

The proof we present is extracted from the proof of Theorem 3 of the Zdun’s paper [

Let

This proposition yields that if

Let

If

The uniqueness part of the theorem follows from Theorems

The following proposition will be used now and later.

Let

The first part of the proposition is in fact Proposition 5 of [

Let

If

Conversely, suppose that

Through this section

If

If

It is clear that

To show that

For every

Let

For each

By this notation one can restate the properties of a Cantor function as follows: If

if

We define the canonical map

The following illustrates general properties of the elements of

Let

if

for every

for each

Put

From

Let

Part (c) is equivalent to saying that for every

A particular subset of

Let

This proposition suggests some notation:

The equivalency of parts (a) and (c) of Proposition

With

Let

This completes the proof.

In general if

Let

Clearly

If

First we show that

To show that

Next we show that

We proceed as follows: (a)

Put

(a)

Now let

(d)

Now we show that

(c)

We now show that

(b)

Let

Let

(a) If

(b) If

(a) First we show that

Now let

(b) By Theorems

Finally, we turn to the map

In this section we use Cantor functions to determine the structure of spoiled disjoint (iteration) subgroups of

Let

Since

Let

the definition of

the definition of

We claim that

Since

We now show that

Next suppose that

We continue with introducing the piecewise linear group generated by a Cantor function. Let

First we must show that the definition of

Clearly

That

To show that

Finally,

Now put

With

First we show that

To show that

Suppose that

To show the maximality of

Let

If

For every Cantor-like set

Now one is ready to determine the structure of spoiled disjoint subgroups of

Let

Let

Suppose that

To show that

Since

This completes the proof.

Let

If

For every Cantor-like set

The next theorem determines the structure of spoiled disjoint iteration subgroups of

Let

Let

Suppose that

This completes the proof.