The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

Mohsen Soltanifar

doi:10.3390/math10050706

Mohsen Soltanifar

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In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces.

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Random fractals have emerged as the natural expansions of deterministic fractals, and their introductory mathematical treatment started with the works of Mandelbrot in the early 1970s [1,2]
The construction process is accomplished in three stages: (i) calculting the cardinality of the power set of a surviving random fractal; (ii) showing the existence of a continuum of random fractals with a plausible fractal dimension and expected Lebesgue measure in n-dimensional Euclidean space Rn(n ≥ 1) and (iii) generalizing the result in the second stage to the cardinal of aleph-two
We begin with the following Lemma of cardinality calculation, which plays a key role in the second generalized Hausdorff dimension theorem (SGHDT)

Summary

Introduction

Random fractals have emerged as the natural expansions of deterministic fractals, and their introductory mathematical treatment started with the works of Mandelbrot in the early 1970s [1,2]. Later, their rigorous mathematical treatment was solidified with the works of Taylor, Falconer and Graf in the mid-1980s [3,4,5]. The small component parts of the fractal have the same probability distribution as the whole fractal Some of their applications have emerged in financial markets [6], cosmogeny [7] and image synthesis [8]. In the virtual random fractals category, include random Cantor sets, random von Koch curves and Galton–Watson fractals

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