[Paper Review] More on one class of fractals

Symon Serbenyuk|arXiv (Cornell University)|Jun 5, 2017

Mathematical Dynamics and Fractals16 references3 citations

TL;DR

This paper investigates fractal sets with Moran structure, focusing on topological, metric, and fractal properties of Cantor-like sets defined by restricted digit or digit-combination usage in s-adic and nega-s-adic representations. It derives exact formulas for Hausdorff dimension using recursive scaling relations and demonstrates that non-self-similar sets (e.g., under nega-s-adic systems) can have strictly higher dimension than their self-similar counterparts.

ABSTRACT

This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.

Motivation & Objective

• To analyze the topological, metric, and fractal properties of sets with Moran structure, particularly those defined by restricted digit usage in s-adic representations.
• To extend the understanding of Hausdorff dimension in non-self-similar fractals, especially those arising from nega-s-adic and alternating Cantor series representations.
• To establish precise dimension formulas for sets where digit combinations are forbidden or constrained, using recursive scaling and measure-theoretic techniques.
• To compare self-similar and non-self-similar fractal sets under the same digit constraints, revealing differences in fractal dimension.
• To contribute to multifractal formalism by analyzing sets with complex scaling behavior not captured by classical self-similar models.

Proposed method

• Uses the Moran construction framework with nested, disjoint, geometrically similar sets to define limit fractal sets in R^n.
• Applies s-adic and nega-s-adic representations to define sets where only specific digit sequences (e.g., {1,2} in ternary) are allowed.
• Derives the Hausdorff dimension via a recursive equation involving scaling factors ω₁, ω₂, ω₃, ω₄ and a sequence (γₖ) satisfying a product identity.
• Employs the Salem function fξ to relate digit-restricted sets to known fractal constructions, enabling dimension transfer.
• Uses Lemmas 6 and 7 to compute infimum and supremum of the sets, establishing bounds for the diameter and local structure.
• Applies Theorem 13 to relate the local structure to the dimension via a limit inferior of γₖ satisfying a multiplicative equation.

Experimental results

Research questions

• RQ1 How does restricting digit combinations in s-adic or nega-s-adic representations affect the Hausdorff dimension of the resulting fractal set?
• RQ2 Can non-self-similar fractals constructed via nega-s-adic systems achieve a higher Hausdorff dimension than their self-similar counterparts with the same digit constraints?
• RQ3 What is the precise formula for the Hausdorff dimension of a Moran-type set defined by forbidden digit patterns in a mixed s-adic system?
• RQ4 How do the scaling factors ω₁, ω₂, ω₃, ω₄ influence the dimension in a recursive, non-uniform scaling framework?
• RQ5 To what extent can the multifractal formalism be extended to sets with complex, non-uniform scaling behavior not captured by classical self-similar models?

Key findings

• For p₀ = 1/6, p₁ = 1/3, p₂ = 1/2, the self-similar set 𝕊_(P₃,0) has a Hausdorff dimension of approximately 0.408985.
• The non-self-similar set 𝕊_(-P₃,0) under the same parameters has a higher Hausdorff dimension of approximately 0.422592.
• When p₀ = p₂ = 0.25 and p₁ = 0.5, both 𝕊_(P₃,0) and 𝕊_(-P₃,0) are self-similar and have a Hausdorff dimension of approximately 0.46496.
• The diameter of the closure of 𝕊_(P₃,0) is d(𝕊̅_(P₃,0)) = p₀ + p₁, while the diameter of its interior is d(𝕊̆_(P₃,0)) = 1 − p₀.
• The local structure of the set is characterized by a recursive branching scheme involving four scaling factors ω₁, ω₂, ω₃, ω₄ with equal distribution of 2ᵏ−² elements at each level k.
• The Hausdorff dimension is given by α* = lim infₖ→∞ γₖ, where (γₖ) satisfies a complex product equation involving the scaling factors and their powers.