- Research Article
- Open Access
Generalized IFSs on Noncompact Spaces
© A. Mihail and R. Miculescu. 2010
- Received: 29 September 2009
- Accepted: 13 January 2010
- Published: 21 January 2010
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.
- Lipschitz Function
- Nonempty Subset
- Hausdorff Dimension
- Natural Generalization
- Unique Fixed Point
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.
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 ). 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  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  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 , 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  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  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.
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 .
which is the same with
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)).
The proofs of the above lemmas are almost obvious.
Concerning the speed of the convergence, one has the following estimation:
See [20, Remark ].
From Theorem 3.7 and Lemma 3.6 we have the following.
Concerning the speed of the convergence, one has the following estimation:
From Theorem 3.11 and Lemma 3.6, we have the following.
From Theorem 3.11, we have
Then, using Lemma 3.2 and Proposition 2.6(ii), we get
Since, according to Lemma 3.6, we have
From Theorem 3.13, Proposition 3.14, and Lemma 3.6, we have the following.
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.
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 .
The authors want to thank the referees whose generous and valuable remarks and comments brought improvements to the paper and enhanced clarity.
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