A Generalization of the Hausdorff Dimension Theorem for Fractals

Abstract

How many fractals exist in nature or the virtual world In this work, we partially answer the second question using Mandelbrots fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of beth-two of virtual fractals with a Hausdorff dimension of a bivariate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

10 pages

References in corpus (2)

• Small-angle scattering from fat fractals
• Structural properties of fractals with positive Lebesgue measures

Cited by in corpus (2)

• The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals
• On the $L^1$ and pointwise divergence of continuous functions