[Paper Review] $\widetilde{Q}$-representation of real numbers and fractal probability distributions

Sergio Albeverio, V. D. Koshmanenko+2 more|ArXiv.org|Aug 1, 2003

Statistical Mechanics and Entropy7 references3 citations

TL;DR

This paper introduces the $\widetilde{Q}$-representation of real numbers as a generalization of $s$-adic expansions, enabling the construction and analysis of fractals and measures with complex local structures. It establishes necessary and sufficient conditions for the distribution of random variables with independent $\widetilde{Q}$-symbols to be absolutely continuous, singular continuous, or pure point, using Hausdorff dimension and symbolic frequency analysis to characterize the fractal properties of their supports.

ABSTRACT

A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $ξ$ with independent $\widetilde{Q}-$symbols are studied in details. Necessary and sufficient conditions for the probability measures $μ_ξ$ associated with $ξ$ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the $\widetilde{Q}-$representation. In addition the metric-topological properties for the distribution of $ξ$ are investigated. A number of examples are presented.

Motivation & Objective

• To develop a generalized symbolic representation ($\widetilde{Q}$-representation) for real numbers that extends $s$-adic expansions and enables the construction of fractals with arbitrary Hausdorff dimension in $[0,1]$.
• To analyze the metric, topological, and fractal properties of probability distributions arising from random variables with independent $\widetilde{Q}$-symbols.
• To classify the resulting probability measures as absolutely continuous, singular continuous, or pure point, based on the $\widetilde{Q}$-representation parameters.
• To determine the Hausdorff-Besicovitch dimension of the distribution of such random variables using symbolic frequency and $\widetilde{Q}$-matrix parameters.

Proposed method

• The $\widetilde{Q}$-representation generalizes $s$-adic expansions by allowing variable digit probabilities and transition rules encoded in a matrix $q_{ik}$.
• Random variables $\xi$ are constructed with independent $\widetilde{Q}$-symbols, where the probability $p_{ik}$ of digit $i$ at position $k$ is specified.
• The support of the distribution is characterized as the set $M[\widetilde{Q},(p_0,\dots,p_{s-1})]$ of points whose $\widetilde{Q}$-representation has asymptotic digit frequencies $p_i$.
• The Hausdorff dimension of the distribution is computed via the formula $\alpha_0(\xi) = \frac{\sum p_i \ln p_i}{\sum p_i \ln q_i}$, derived from symbolic dynamics and measure-theoretic analysis.
• Topological and metric properties of the support are analyzed using the essential support $N_\xi^\infty$, which captures the minimal set of full measure.
• The paper uses the decomposition of singular continuous measures into pure $C$-, $S$-, and $P$-type components to classify the resulting distributions.

Experimental results

Research questions

• RQ1 What conditions on the $\widetilde{Q}$-representation and symbol probabilities $p_{ik}$ ensure that the distribution of $\xi$ is absolutely continuous?
• RQ2 When is the distribution of $\xi$ singular continuous or pure point, and how do these types relate to the topological and metric structure of the support?
• RQ3 How can the Hausdorff-Besicovitch dimension of the distribution of $\xi$ be computed from the $\widetilde{Q}$-matrix and symbol probabilities?
• RQ4 What is the relationship between the essential support $N_\xi^\infty$ and the fractal structure of the distribution, especially in singular cases?
• RQ5 Can the $\widetilde{Q}$-representation generate fractals with any prescribed Hausdorff dimension $\alpha_0 \in [0,1]$?

Key findings

• The distribution of $\xi$ is absolutely continuous if the symbol probabilities $p_{ik}$ are constant and symmetric, e.g., $p_{0k} = p_{2k} = 1/2$, $p_{1k} = 0$, yielding a Lebesgue-absolutely continuous measure of pure $P$-type.
• The distribution is pure point if $p_{0k} = 1 - 1/2^k$, $p_{1k} = 0$, $p_{2k} = 1/2^k$, resulting in a discrete measure supported on a countable set of Hausdorff dimension zero.
• The distribution is singular continuous if $p_{0k} = 1/4$, $p_{1k} = 0$, $p_{2k} = 3/4$, yielding a measure of pure $P$-type with support of positive Hausdorff dimension but zero Lebesgue measure.
• The Hausdorff-Besicovitch dimension of the distribution is given by $\alpha_0(\xi) = \frac{\sum p_i \ln p_i}{\sum p_i \ln q_i}$, where $p_i$ and $q_i$ are the asymptotic frequencies and scaling factors of the $\widetilde{Q}$-representation.
• The essential support $N_\xi^\infty$ is a more informative descriptor than the topological support for singular distributions, as it captures the minimal set of full measure with complex local structure.
• The $\widetilde{Q}$-representation allows the construction of everywhere dense, noncompact fractals with any desired Hausdorff dimension $\alpha_0 \in [0,1]$.