Generalized IFSs on Noncompact Spaces

Abstract

The aim of this paper is to continue the research work that we have done in a previous paper published in this journal (see Mihail and Miculescu, 2008). We introduce the notion of GIFS, which is a family of functions , where is a complete metric space (in the above mentioned paper the case when is a compact metric space was studied) and . In case that the functions are Lipschitz contractions, we prove the existence of the attractor of such a GIFS and explore its properties (among them we give an upper bound for the Hausdorff-Pompeiu distance between the attractors of two such GIFSs, an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of and we prove its continuous dependence in the 's). Finally we present some examples of attractors of GIFSs. The last example shows that the notion of GIFS is a natural generalization of the notion of IFS.

1. Introduction

1.1. The Organization of the Paper

The paper is organized as follows. Section 2 contains a short presentation of the notion of an iterated function system (IFS), one of the most common and most general ways to generate fractals. This will serve as a framework for our generalization of an iterated function system.

Then, we introduce the notion of a GIFS, which is a finite family of Lipschitz contractions , where is a complete metric space and .

In Section 3 we prove the existence of the attractor of such a GIFS and explore its properties (among them we give an upper bound for the Hausdorff-Pompeiu distance between the attractors of two such GIFSs, an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of and we prove its continuous dependence in the 's).

Section 4, the last one, contains some examples and remarks. The last example shows that the notion of GIFS is a natural generalization of the notion of IFS.

1.2. Some Generalizations of the Notion of IFS

IFSs were introduced in their present form by John Hutchinson and popularized by Barnsley (see [1]). There is a current effort to extend Hutchinson's classical framework for fractals to more general spaces and infinite IFSs. Some papers containing results on this direction are [2–7].

1.3. Some Physical Applications of IFSs

In the last period IFSs have attracted much attention being used by researchers who work on autoregressive time series, engineer sciences, physics, and so forth. For applications of IFSs in image processing theory, in the theory of stochastic growth models, and in the theory of random dynamical systems one can consult [8–10]. Concerning the physical applications of iterated function systems we should mention the seminal paper [11] of El Naschie which draws attention to an informal but instructive analogy between iterated function systems and the two-slit experiment which is quite valuable in illuminating the role played by the possibly DNA-like Cantorian nature of microspacetime and clarifies the way in which probability enters into this subject. We also mention the paper [12] of Słomczyński where a new definition of quantum entropy is introduced and one method (using the theory of iterated function systems) of calculating coherent states entropy is presented. The coherent states entropy is computed as the integral of the Boltzmann-Shannon entropy over a fractal set.

In [13], Bahar described bifurcation from a fixed-point generated by an iterated function system (IFS) as well as the generation of "chaotic" orbits by an IFS, and in [14] unusual and quite interesting patterns of bifurcation from a fixed-point in an IFS system, as well as the routes to chaos taken by IFS-generated orbits, are discussed. Moreover, in [15] it is shown that random selection of transformation in the IFS is essential for the generation of a chaotic attractor. In [16, Section ], one can find a lengthy but elementary explanation which features of randomness play the main role.

2. Preliminaries

Notations 2.

Let and be two metric spaces.

As usual, denotes the set of continuous functions from to , and defined by

(2.1)

is the generalized metric on .

For a sequence of elements of and , denotes the punctual convergence, denotes the uniform convergence on compact sets, and denotes the uniform convergence, that is, the convergence in the generalized metric .

Definition 2.1.

Let be a complete metric space and .

For a function , the number

(2.2)

which is the same with

(2.3)

is denoted by and it is called the Lipschitz constant of .

A function is called a Lipschitz function if and a Lipschitz contraction if .

We will use the notation for the set .

Notations 2.

denotes the subsets of a given set and denotes the set .

For a subset of , by we mean .

Given a metric space , denotes the set of compact subsets of and denotes the set of closed bounded subsets of .

Remark 2.2.

It is obvious that .

Definition 2.3.

For a metric space , we consider on the generalized Hausdorff-Pompeiu pseudometric defined by and , where and .

Remark 2.4.

The Hausdorff-Pompeiu pseudometric is a metric on and, in particular, on .

Remark 2.5.

The metric spaces and are complete, provided that is a complete metric space (see [1, 7, 17]).

The following proposition contains the important properties of the Hausdorff-Pompeiu semimetric (see [1, 17] or [18]).

Proposition 2.6.

Let and be two metric spaces. Then one has the following:

(i)if and are two nonempty subsets of , then

(2.4)

(ii)if and are two families of nonempty subsets of , then

(2.5)

(iii)if and are two nonempty subsets of and is a Lipschitz function, then

(2.6)

Definition 2.7.

An iterated function system on consists of a finite family of Lipschitz contractions on and is denoted .

Theorem 2.8.

Let be a complete metric space, let be an IFS. Then there exists a unique such that

(2.7)

The set is called the attractor of the IFS .

Given a metric space , the idea of our generalization of the notion of an IFS is to consider contractions from to , rather than contractions from to itself.

Definition 2.9.

Let be a complete metric space and . A generalized iterated function system on of order (for short a GIFS or a GmIFS), denoted , consists of a finite family of functions , such that .

Earlier several authors tried to coin the name generalized IFS. One should note the paper [19] in which notion tightly corresponds to contractive multivalued IFS from [2].

3. The Existence of the Attractor of a GIFS for Lipschitz Contractions

In this section is a fixed natural number, will be a fixed complete metric space, and all the GIFSs are of order and have the form , where is a natural number.

We prove the existence of the attractor of (Theorem 3.9) and study its properties (among them we give an upper bound for the Hausdorff-Pompeiu distance between the attractors of two such GIFSs (Theorem 3.12), an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of (Theorem 3.17) and we prove its continuous dependence in the 's (Theorem 3.15)).

Definition 3.1.

Let be a function. The function defined by

(3.1)

for all , is called the set function associated to the function .

The function defined by

(3.2)

for all , is called the set function associated to the GIFS .

Lemma 3.2.

For a sequence of elements of and such that , one has

(3.3)

in, for all.

Proposition 3.3.

Let and be two complete metric spaces and let be such that and on a dense set in . Then

(3.4)

Proof.

In this proof, by we mean .

Let us consider , which is a dense set in , let be a compact set in , and let .

Since is uniformly continuous on , there exists such that if and , then

(3.5)

Since is compact, there exist such that

(3.6)

Taking into account the fact that is dense in , we can choose such that

(3.7)

Since, for all , , there exists such that for every , , we have

(3.8)

for every .

For , there exists , such that and therefore

(3.9)

and so

(3.10)

Hence, for , we have

(3.11)

Consequently, as was arbitrary chosen in , we infer that on , and so

(3.12)

The inequality

(3.13)

is obvious.

Lemma 3.4.

Let be subsets of  .

Then

(1);

(2).

Lemma 3.5.

If is a Lipschitz function, then

(3.14)

Lemma 3.6.

In the framework of this section, one has

(3.15)

The proofs of the above lemmas are almost obvious.

Theorem 3.7 (Banach contraction principle for ).

For every , there exists a unique , such that

(3.16)

For every , the sequence , defined by , for all , has the property that

(3.17)

Concerning the speed of the convergence, one has the following estimation:

(3.18)

for every .

Proof.

See [20, Remark ].

Remark 3.8.

The point from the above theorem is called the fixed point of .

From Theorem 3.7 and Lemma 3.6 we have the following.

Theorem 3.9.

In the framework of this section, there exists a unique such that

(3.19)

Moreover, for any , the sequence defined by , for all , has the property that

(3.20)

Concerning the speed of the convergence, one has the following estimation:

(3.21)

for all .

Definition 3.10.

The unique set given by the previous theorem is called the attractor of the GIFS .

Theorem 3.11.

If have the fixed points and , then

(3.22)

Proof.

We have

(3.23)

so

(3.24)

and in a similar manner we get

(3.25)

Therefore

(3.26)

From Theorem 3.11 and Lemma 3.6, we have the following.

Theorem 3.12.

In the framework of this section, if andare two dimensional GIFSs, then

(3.27)

where  .

Theorem 3.13.

Let with fixed points and , respectively, such that

(3.28)

on a dense set in .

Then

(3.29)

Proof.

From the fact that and on a dense set in , it follows, using Proposition 3.3, that

(3.30)

on and

(3.31)

From Theorem 3.11, we have

(3.32)

and hence

(3.33)

for all .

Since on , we obtain that

(3.34)

and consequently, using the above inequality, we obtain that

(3.35)

Proposition 3.14.

Let , where , and let be -dimensional generalized iterated function systems such that

(3.36)

on a dense set in , for every .

Then

(3.37)

Proof.

Using Proposition 3.3, we obtain that

(3.38)

on and

(3.39)

Then, using Lemma 3.2 and Proposition 2.6(ii), we get

(3.40)

Since, according to Lemma 3.6, we have

(3.41)

for all , we obtain, using again the arguments from Proposition 3.3, that

(3.42)

From Theorem 3.13, Proposition 3.14, and Lemma 3.6, we have the following.

Theorem 3.15.

Let , where , and let be -dimensional generalized iterated function systems having the property that

(3.43)

on a dense set in , for every .

Then

(3.44)

Theorem 3.16.

For having the unique fixed point and for every , one has

(3.45)

Proof.

We can use the Banach contraction principle for , where

(3.46)

for all .

Theorem 3.17.

For a generalized iterated function system and , the following inequality is valid:

(3.47)

Proof.

The function , defined by

(3.48)

for all , is a contraction and

(3.49)

4. Examples

In this section we present some examples of attractors of GIFSs. Example 4.3 shows that the notion of GIFS is a natural generalization of the notion of IFS.

Example 4.1.

Let and , where is a Banach space and is the set of linear and continuous operators from to .

Let us consider the function , given by

(4.1)

for every .

Then

(4.2)

for every , and so

(4.3)

In particular, if and , for every , then

(4.4)

Let us consider given by

(4.5)

for every .

Then

(4.6)

If , then are contractions.

We consider the GIFS, where.

If , then

(4.7)

Indeed, , and so , that is, . This proves that .

If , then

(4.8)

Indeed, if , then

(4.9)

and . Hence . This proves that .

If , then

(4.10)

If , then is a Cantor type set (more precisely consists of those elements of for which one can use the digits and in order to write them in base ).

Remark 4.2.

Finally let us note that

(4.11)

Example 4.3.

Let be one of the spaces , or , where .

Let , and be given by

(4.12)

for all .

We consider the GIFS , where and are given by

(4.13)

for all .

Then

(4.14)

Indeed, if , then and . Hence

(4.15)

and therefore . This, together with the fact that is compact, proves that .

On one hand it is obvious that has infinite Hausdorff dimension. On the other hand, for every finite IFS , with contraction constant less then , we have . Indeed, the proof of the above claim is similar with the one of Proposition , page 135, from [18].

Therefore there exists no finite IFS consisting of Lipschitz contractions having as attractor the set .