Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Self-affine sponges with random contractions

Balázs Bárány, Antti Käenmäki, Michał Rams

TL;DR

The paper proves that random self-affine sponges without separation conditions have almost surely full dimension given by $\dim_H(X)=\dim_M(X)=\min\{d, s_0\}$, where $s_0$ is determined by a pressure equation involving the random contraction matrices. The authors develop a martingale-based construction of a random measure on the symbolic space, establish transversality for both line projections and full $d$-dimensional sponges, and combine these with dimension upper/lower bound techniques to obtain the main result. When $s_0>d$, the sponge has positive Lebesgue measure, as shown via absolute continuity of the push-forward measure. The work extends Koivusalo’s dimension results to higher dimensions under random contractions and provides concrete examples (random line sets and random 4-corner sponges) illustrating the theory and dimension values. This advances understanding of fractal dimensions in random affine systems without separation assumptions and highlights the role of probabilistic transversality in achieving sharp almost-sure dimension formulas.

Abstract

We compute the almost sure Hausdorff dimension of random self-affine sponges in $\mathbb{R}^d$ without imposing any separation conditions. In this context, randomness arises from the matrices in the defining semigroup, which are random yet the corresponding affine maps share a fixed point.

Self-affine sponges with random contractions

TL;DR

The paper proves that random self-affine sponges without separation conditions have almost surely full dimension given by , where is determined by a pressure equation involving the random contraction matrices. The authors develop a martingale-based construction of a random measure on the symbolic space, establish transversality for both line projections and full -dimensional sponges, and combine these with dimension upper/lower bound techniques to obtain the main result. When , the sponge has positive Lebesgue measure, as shown via absolute continuity of the push-forward measure. The work extends Koivusalo’s dimension results to higher dimensions under random contractions and provides concrete examples (random line sets and random 4-corner sponges) illustrating the theory and dimension values. This advances understanding of fractal dimensions in random affine systems without separation assumptions and highlights the role of probabilistic transversality in achieving sharp almost-sure dimension formulas.

Abstract

We compute the almost sure Hausdorff dimension of random self-affine sponges in without imposing any separation conditions. In this context, randomness arises from the matrices in the defining semigroup, which are random yet the corresponding affine maps share a fixed point.
Paper Structure (11 sections, 10 theorems, 108 equations, 3 figures)

This paper contains 11 sections, 10 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction and results
  2. Preliminaries
  3. Main result and examples
  4. Construction of random measures
  5. Transversality over subsystems
  6. Transversality on the line
  7. Transversality for random self-affine sponges
  8. Dimension of random self-affine sponges
  9. Dimension upper bound
  10. Dimension lower bound
  11. Positive Lebesgue measure

Key Result

Theorem 1.1

If $X \subset \mathbb{R}^d$ is a self-affine sponge whose coordinate projections satisfy the exponential separation condition and has a simple Lyapunov spectrum, then

Figures (3)

  • Figure 1: Illustration for the self-similar set with random contractions defined in \ref{['ex:first']}.
  • Figure 2: The first and second iterates of the RIFS defined in \ref{['eq:rifs2']}, and the random 4-corner set $X$ of \ref{['ex:4corner']}.
  • Figure 3: The first and second iterates of the RIFS defined in \ref{['eq:rifs2b']}, and the random 4-corner set $X$ of \ref{['ex:mod4corner']}.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 11 more