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Quasisymmetric mappings on two variants of fractal percolation

Roope Anttila, Sylvester Eriksson-Bique, Aleksi Pyörälä

TL;DR

The paper investigates how quasisymmetric mappings interact with the conformal dimension of two random fractal percolation variants—fat and dense. It develops deterministic Cantor-set tools and a branching-process framework to show that fat fractal percolation is almost surely minimal for the conformal Hausdorff dimension conditioned on non-extinction, i.e., $\mathcal{C}dim_H F = \dim_H F = d$. For dense fractal percolation, the authors prove that the Hausdorff dimension cannot be lowered by power quasisymmetries, i.e., $\dim_H f(E) = \dim_H E = d$ for any power quasisymmetry $f$, though full minimality under all quasisymmetries remains open. The results extend earlier work on random fractals with Galton–Watson structure by handling non-uniform offspring via stepwise-probability changes and varying subdivision counts, highlighting how large-scale holes and deterministic substructures govern conformal-invariant dimensions. The methods have implications for understanding quasisymmetric invariants in random fractals and contribute to the broader theory of conformal dimension in metric spaces with randomized constructions.

Abstract

We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.

Quasisymmetric mappings on two variants of fractal percolation

TL;DR

The paper investigates how quasisymmetric mappings interact with the conformal dimension of two random fractal percolation variants—fat and dense. It develops deterministic Cantor-set tools and a branching-process framework to show that fat fractal percolation is almost surely minimal for the conformal Hausdorff dimension conditioned on non-extinction, i.e., . For dense fractal percolation, the authors prove that the Hausdorff dimension cannot be lowered by power quasisymmetries, i.e., for any power quasisymmetry , though full minimality under all quasisymmetries remains open. The results extend earlier work on random fractals with Galton–Watson structure by handling non-uniform offspring via stepwise-probability changes and varying subdivision counts, highlighting how large-scale holes and deterministic substructures govern conformal-invariant dimensions. The methods have implications for understanding quasisymmetric invariants in random fractals and contribute to the broader theory of conformal dimension in metric spaces with randomized constructions.

Abstract

We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-extinction, the Hausdorff dimension of the fat fractal percolation cannot be lowered with a quasisymmetry and the Hausdorff dimension of the dense fractal percolation cannot be lowered with a power quasisymmetry.

Paper Structure

This paper contains 13 sections, 12 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conformal dimension
  3. Conformal dimension of random fractals
  4. Fat and dense fractal percolations
  5. Fat fractal percolation
  6. Dense fractal percolation
  7. On the proofs
  8. Notation
  9. Fat and dense Cantor sets
  10. Fat Cantor sets
  11. Dense Cantor sets
  12. Branching processes
  13. Proofs of Theorems \ref{['thm:main1']} and \ref{['thm:main2']}

Key Result

Theorem 2.1

Let $N\geq 2$. For any sequence $\textbf{p} = (p_n)_{n\in\mathbb{N}}$ such that $\lim_{n\to\infty} p_n = 1$, we have almost surely conditioned on the non-extinction of $F = F(N, \textbf{p})$.

Figures (1)

  • Figure 1: On the left a realization of the $(3,0.5)$-fractal percolation, in the middle, a realization of the $(3, \mathbf{p})$-fat fractal percolation with $p_0 = 0.5$, $p_1 = 0.6$, $p_2 = 0.65,\ldots$, and on the right a realization of the $(\mathbf{N},0.5)$-dense fractal percolation with $N_1=3$, $N_2=6$, $N_3=9,\ldots$.

Theorems & Definitions (20)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • Lemma 3.6
  • ...and 10 more