On different forms of self similarity
Introduction
Fractals and self similarity are two intense areas of current mathematical research. Fractals allow us to see the world as it is. Roughly fractal theory is the study of shapes of natural objects. Self similarity is a significant property of fractals. There are different forms of self similarity in mathematics and nature. They include super, sub, partial and quasi self similar forms.
Fractals were introduced and studied by Mandelbrot [3] for the first time in 1975. Then many mathematicians including Barnsley[15] and Falconer [6], [7], [8], [9], [10], [11], [12], [13] carried the research forward. Self similarity was mathematically defined by Hutchinson [5] in 1981 and the concepts of sub and super self similarities were introduced by Falconer[9] in 1995 and subsequently studied by Bandt [4], Schief [1], [2], Mcclure and Vallin[16] and many others. More related works can be seen in [17], [18], [19], [21], [23].
Abstract spaces are abundant in mathematics, physics and other areas of science and technology. Even though it is impossible to visualize functions, surfaces and different types of sets in these spaces, they do exist in some complicated mathematical form. For example, in physics a ‘state space’ is an abstract space in which different positions represent different states of the physical system. In quantum mechanics theory, a state space is a complex Hilbert space. Also, the four dimensional ‘space time’ may have some fractal properties on a quantum scale. Fractals exist in abstract Sobolov spaces as self similar energy forms [20]. It is possible that self similar fractals exist even in the sub atomic levels.
Even though the complete structure of fractals in abstract spaces are difficult to obtain, some of their properties like fractal dimension can be studied and conclusions can be obtained. It is shown in [14] that the typical trajectories of quantum mechanical particles are continuous but not differentiable. But they can be characterized by their fractal dimension.
Having these in mind, some of the existing results in ℜn (Corollary 2.2[16] and Theorem 3.3[16]) are generalized to arbitrary spaces. Also, for the mathematical completion, our definitions will be given in abstract metric spaces. But all the examples will be in ℜn, particularly in ℜ2, for a better understanding.
Even though all forms of self similarity are present in nature, super self similarity is the most common and abundant one. Most of the objects in nature like trees, ferns, flowers etc., are super self similar. In this paper, we look super self similar sets more closely and discuss some of their properties. Also, some new forms of self similarity are introduced and related results are discussed.
Section snippets
Preliminaries
In this section, some of the basic definitions and results required for the development of this paper are presented. The concepts of Hausdorff metric space, Hausdorff distance, contractions and similarities are discussed in detail. The fixed point of a contraction in a complete metric space can be determined by means of contraction mapping theorem, which is also given in this section. We shall denote (X, d), a complete metric space and (H(X), h), the corresponding Hausdorff metric space.
Definition 2.1 Let ([15]
Main definitions and results
Super self similarity differs from self and sub self similarity in many ways.
Uniqueness of super self similar sets: Unlike self similar sets, super self similar sets with respect to a set of contracting similarities need not be unique. For example, consider the contracting similarities and in the real line. Both the intervals [0, 1] and are super self similar with respect to S1 and S2.
Union of super self similar sets: In [9] Falconer showed that union of two sub self
Conclusion
Different forms of self similarities are discussed in this paper. They include, super, sub, partial and exact self similarities. Analytical and topological properties of sets exhibiting these forms of self similarities are studied. It is proved that a set is super self similar if and only if each of the contractions associated with the set are contracting sub similarities. Certain new forms of self similarity, such as partial super self similarity and, exact sub and super self similarities are
Acknowledgement
The authors gratefully acknowledge the many helpful suggestions and stimulating conversations of Professor Kenneth J Falconer and Professor Cristoph Bandt during the preparation of this paper.
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