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The dimension of random subsets of self-similar sets generated by branching random walk

Pieter Allaart, Lauritz Streck

Abstract

Given a self-similar set $Λ$ that is the attractor of an iterated function system (IFS) $\{f_1,\dots,f_N\}$, consider the following method for constructing a random subset of $Λ$: Let $\mathbf{p}=(p_1,\dots,p_N)$ be a probability vector, and label all edges of a full $M$-ary tree independently at random with a number from $\{1,2,\dots,N\}$ according to $\mathbf{p}$, where $M\geq 2$ is an arbitrary integer. Then each infinite path in the tree starting from the root receives a random label sequence which is the coding of a point in $Λ$. We let $F\subsetΛ$ denote the set of all points obtained in this way. This construction was introduced by Allaart and Jones [J. Fractal Geom. 12 (2025), 67--92], who considered the case of a homogeneous IFS on $\mathbb{R}$ satisfying the Open Set Condition (OSC) and proved non-trivial upper and lower bounds for the Hausdorff dimension of $F$. We demonstrate that under the OSC, the Hausdorff (and box-counting) dimension of $F$ is equal to the upper bound of Allaart and Jones, and extend the result to higher dimensions as well as to non-homogeneous self-similar sets.

The dimension of random subsets of self-similar sets generated by branching random walk

Abstract

Given a self-similar set that is the attractor of an iterated function system (IFS) , consider the following method for constructing a random subset of : Let be a probability vector, and label all edges of a full -ary tree independently at random with a number from according to , where is an arbitrary integer. Then each infinite path in the tree starting from the root receives a random label sequence which is the coding of a point in . We let denote the set of all points obtained in this way. This construction was introduced by Allaart and Jones [J. Fractal Geom. 12 (2025), 67--92], who considered the case of a homogeneous IFS on satisfying the Open Set Condition (OSC) and proved non-trivial upper and lower bounds for the Hausdorff dimension of . We demonstrate that under the OSC, the Hausdorff (and box-counting) dimension of is equal to the upper bound of Allaart and Jones, and extend the result to higher dimensions as well as to non-homogeneous self-similar sets.

Paper Structure

This paper contains 9 sections, 13 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Connections with the literature
  3. Definitions and statements of main results
  4. Sketch of proof of Theorem \ref{['thm:variational']}
  5. The random stopping set
  6. Proof of the upper bound
  7. Proof of the lower bound
  8. Solving the constrained optimization
  9. Proofs of the special cases

Key Result

Theorem 1.1

For the homogeneous IFS $\{f_1,f_2\}$ with equal ratios $r_1=r_2=r$ and the binary tree, almost surely, where $\xi=\xi(p)$ is defined by for $p \neq 1/2$, and $\xi(1/2):=1/2$.

Figures (1)

  • Figure 1: The graph of $\dim_H F$ as a function of $p$ in Theorem \ref{['cor:two-maps-nonhomogeneous']}, for $r_1=0.2$ and $r_2=0.7$. Here $s_0\approx 0.8398$, $p_*\approx 0.0900$ and $p^*\approx 0.7903$. The dashed curve between $p_*$ and $p^*$ is the continuation of the graph of $\hat{s}(p)$ where it lies strictly below $\dim_H F$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4: The homogeneous case
  • Remark 2.5
  • Remark 2.6
  • Proposition 4.1
  • ...and 18 more