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Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

Jan Rataj, Steffen Winter, Martina Zähle

TL;DR

The paper develops a mean-curve framework for random self-similar code-tree fractals, establishing existence and explicit integral representations for mean Lipschitz-Killing curvatures of parallel sets as the radius $\varepsilon$ shrinks, under Uniform Strong Open Set Conditions and geometric regularity. By embedding the fractals in a flexible code-tree model with back-path independence, it unifies random recursive, homogeneous, and $V$-variable constructions and extends known deterministic and random results. The main contributions include renewal-theoretic expressions for the limits, a mean Minkowski-content result, and a discussion of when the Minkowski dimension in the mean may differ from the almost-sure dimension, with implications for the geometry and connectivity of random fractals. The framework accommodates dependencies within and across levels and provides a pathway to analyze a broad class of random fractals beyond classical models.

Abstract

We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions.

Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

TL;DR

The paper develops a mean-curve framework for random self-similar code-tree fractals, establishing existence and explicit integral representations for mean Lipschitz-Killing curvatures of parallel sets as the radius shrinks, under Uniform Strong Open Set Conditions and geometric regularity. By embedding the fractals in a flexible code-tree model with back-path independence, it unifies random recursive, homogeneous, and -variable constructions and extends known deterministic and random results. The main contributions include renewal-theoretic expressions for the limits, a mean Minkowski-content result, and a discussion of when the Minkowski dimension in the mean may differ from the almost-sure dimension, with implications for the geometry and connectivity of random fractals. The framework accommodates dependencies within and across levels and provides a pathway to analyze a broad class of random fractals beyond classical models.

Abstract

We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions.
Paper Structure (6 sections, 8 theorems, 104 equations, 3 figures, 1 table)

This paper contains 6 sections, 8 theorems, 104 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of random self-similar code tree fractals
  4. Examples
  5. Markov stops and statement of the main results
  6. Proofs

Key Result

Lemma 2.1

For any $R>\sqrt{2}$ and $k=0,1,\ldots,d$ there exists a constant $c_k(R)$ such that for any compact set $A\subset\mathbb{R}^d$ and any $r\ge R|A|$, $\partial (A(r))$ is a $(d-1)$-dimensional Lipschitz submanifold, and

Figures (3)

  • Figure 1: Illustration of the first construction step of the two IFS $G$ and $G'$ in Example \ref{['ex:dependent-SG']} used to generate a random Sierpiński gasket with dependencies within the levels.
  • Figure 2: Realization of the the first three construction steps of the random Sierpiński gasket in Example \ref{['ex:dependent-SG']}. The brown color indicates the 'exceptional' corner at each step (determined by the number $M_n$).
  • Figure 3: The mappings $g_1,\ldots, g_4$ used in the IFSs $G_1,\ldots G_4$ in Example \ref{['ex:percolation']}. Depicted are the images $g_i(Q)$ of the unit square $Q$ (left) and the unions of the images of $Q$ under the mappings of the IFSs $G_1,\ldots, G_4$ (right).

Theorems & Definitions (22)

  • Lemma 2.1: Za11
  • Lemma 2.2: RWZ23
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Proposition 5.1
  • proof
  • ...and 12 more