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Properties of a random Cantor set with overlaps

Anna Chiara Lai, Paola Loreti

TL;DR

The paper analyzes a random Cantor set with overlaps generated by an IFS based on the golden mean in one dimension, where the Open Set Condition fails. It employs expansions in non-integer bases and greedy representations to quantify overlaps and the surviving-interval count, culminating in a closed-form expression for the expected Hausdorff dimension: $\mathbb{E}(\dim(\mathcal{C}^p))=0$ for $p\in[0,1/2]$, $\log(2p)/\log \varphi$ for $p\in[1/2,\varphi/2]$, and $1$ for $p\in[\varphi/2,1]$, with a novel non-OSC threshold $p_d=\varphi/2$. The results extend OSC-based dimension formulas to overlap-rich settings and reveal a percolation-type threshold phenomenon in the fractal topology, suggesting avenues for higher-dimensional extensions and bases $q\in(1,\varphi)$. The work bridges fractal geometry with non-integer base representations, providing a combinatorial framework to count exact overlaps and quantify density phenomena in random Cantor sets.

Abstract

We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set Condition fails. Our methodology is based on the theory of expansions in non-integer bases.

Properties of a random Cantor set with overlaps

TL;DR

The paper analyzes a random Cantor set with overlaps generated by an IFS based on the golden mean in one dimension, where the Open Set Condition fails. It employs expansions in non-integer bases and greedy representations to quantify overlaps and the surviving-interval count, culminating in a closed-form expression for the expected Hausdorff dimension: for , for , and for , with a novel non-OSC threshold . The results extend OSC-based dimension formulas to overlap-rich settings and reveal a percolation-type threshold phenomenon in the fractal topology, suggesting avenues for higher-dimensional extensions and bases . The work bridges fractal geometry with non-integer base representations, providing a combinatorial framework to count exact overlaps and quantify density phenomena in random Cantor sets.

Abstract

We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set Condition fails. Our methodology is based on the theory of expansions in non-integer bases.
Paper Structure (5 sections, 3 theorems, 58 equations, 2 figures)

This paper contains 5 sections, 3 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A random Cantor set with overlaps and its properties
  3. Critical probabilities
  4. Conclusions and perspectives
  5. Proof of Proposition \ref{['p1']}

Key Result

Proposition 1

Let $n\geq 0$ and consider the collection $\mathcal{I}_n=\{I_\mathbf c=f_\mathbf c(I_0)\mid \mathbf c\in\{0,1\}^n\}$. Then

Figures (2)

  • Figure 1: Band visualization of the deterministic case. Each horizontal band corresponds to the union of the intervals in $\mathcal{I}_n$, for $n=0,\dots,6$. Interval intersections are highlighted by shading. For instance the third band corresponds to $\mathcal{I}_2=\{[0,1/\varphi],[1/\varphi,2/\varphi],[1/\varphi^2,1],[1,\varphi]\}$, the darker central area is the union of the pairwise, disjoint intersections $[0,1/\varphi]\cap [1/\varphi^2,1]$, $[1/\varphi,2/\varphi]\cap [1/\varphi,1]$ and $[1/\varphi^2,1]\cap [1/\varphi,2/\varphi]$.
  • Figure 2: Band visualization of a random prefractal with $p=0.7$. Each horizontal band corresponds to $\mathcal{C}^p_n$ for $n=0,\dots,10$. Interval intersections are highlighted by shading.

Theorems & Definitions (7)

  • Example 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof