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The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces

Riku Anttila, Sylvester Eriksson-Bique

TL;DR

This work develops a general framework to construct symmetric self-similar metric spaces as limits of vertex-Iterated Graph Systems, unifying classical fractals (like the Sierpiński carpet and Menger sponges) with new examples (pillow space, pentagonal carpet). It introduces a replacement-flow method to translate flows on base graphs into flows on larger replacements, enabling robust modulus estimates and the key super-multiplicativity property for discrete p-moduli, via the duality with p-resistance. The main contributions prove that these limit spaces carry the Combinatorial Loewner Property with a conformal-dimension exponent Q, establishing a precise link between discrete moduli and conformal geometry in a broad self-similar setting. The results open pathways to defining Sobolev spaces, Dirichlet forms, and diffusion processes on these fractal spaces, bridging discrete graph techniques with continuum potential theory and quasisymmetric uniformization.

Abstract

This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carpet and the higher dimensional Menger sponges, but also a rich class of new examples. The main result of the work roughly speaking states: If the construction is sufficiently symmetric then the limiting object supports useful moduli estimates, namely the Combinatorial Loewner property of Bourdon--Kleiner and the super-multiplicativity inequalities. The latter are established on Menger sponges for which it had not been previously known. The main new technique the work offers is a general framework of flows and resistance estimates.

The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces

TL;DR

This work develops a general framework to construct symmetric self-similar metric spaces as limits of vertex-Iterated Graph Systems, unifying classical fractals (like the Sierpiński carpet and Menger sponges) with new examples (pillow space, pentagonal carpet). It introduces a replacement-flow method to translate flows on base graphs into flows on larger replacements, enabling robust modulus estimates and the key super-multiplicativity property for discrete p-moduli, via the duality with p-resistance. The main contributions prove that these limit spaces carry the Combinatorial Loewner Property with a conformal-dimension exponent Q, establishing a precise link between discrete moduli and conformal geometry in a broad self-similar setting. The results open pathways to defining Sobolev spaces, Dirichlet forms, and diffusion processes on these fractal spaces, bridging discrete graph techniques with continuum potential theory and quasisymmetric uniformization.

Abstract

This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carpet and the higher dimensional Menger sponges, but also a rich class of new examples. The main result of the work roughly speaking states: If the construction is sufficiently symmetric then the limiting object supports useful moduli estimates, namely the Combinatorial Loewner property of Bourdon--Kleiner and the super-multiplicativity inequalities. The latter are established on Menger sponges for which it had not been previously known. The main new technique the work offers is a general framework of flows and resistance estimates.
Paper Structure (31 sections, 51 theorems, 280 equations, 11 figures)

This paper contains 31 sections, 51 theorems, 280 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Basic framework
  4. Main results
  5. Main techniques
  6. Preliminaries
  7. Conformal dimension
  8. Terminology of graphs
  9. Graph approximations
  10. Iterated graph systems
  11. Definitions and notations
  12. Replacement rule
  13. Symmetric IGSs
  14. Main Examples
  15. Hypercube based construction
  16. ...and 16 more sections

Key Result

Theorem 1.5

Let $(X,d)$ be a $d$-dimensional Menger sponge for $d \geq 3$, the pentagonal carpet, the pillow space or more generally any metric space satisfying the assumptions in Theorem thm:supermult. For $p \geq 1$ let $\mathop{\mathrm{\mathcal{M}}}\nolimits_{p,n}$ be as we defined above. Then the following The constant $C$ depends on the ambient space and continuously on $p \in [1,\infty)$.

Figures (11)

  • Figure 1: Construction of the Sierpiński carpet as a limit of graphs.
  • Figure 2: First three steps in the construction of the pentagonal carpet
  • Figure 3: First two steps in the construction of the pillow space.
  • Figure 4: Figure of a symmetric hypercubic IGS. The limit space can also be obtained by adding horizontal- and vertical "slices" to a unit square in a self-similar fashion.
  • Figure 5: In this figure we point out that our definition of hypercubic IGS does not allow direct diagonal connections. The construction in the figure is very similar to an Unconstrained Sierpiński carpet in cao2024whether. But it is nevertheless very different because the fractal space in the figure does not contain local cut-points.
  • ...and 6 more figures

Theorems & Definitions (136)

  • Example 1.1: Sierpiński carpet
  • Example 1.2: Pentagonal carpet
  • Example 1.2: Pentagonal carpet
  • Example 1.3: Pillow space
  • Theorem 1.5
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 126 more