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Universal Differentiability Sets in Laakso Space

Sylvester Eriksson-Bique, Andrea Pinamonti, Gareth Speight

TL;DR

This work proves that Laakso space $F$ admits a continuous family of mutually singular doubling measures $\{\mu_w\}_{w\in(0,1)}$ for which every Lipschitz function is differentiable almost everywhere with respect to each $\mu_w$, thereby producing measure-zero universal differentiability sets in $F$. The authors construct $\nu_w$ on the Cantor component and push forward $\mathcal{H}^1\times \nu_w$ to obtain $\mu_w$, with the corresponding sets $N_w$ of full $\mu_w$-measure whose complements have Hausdorff measure zero and which are mutually singular for distinct $w$. A key feature is that each $(F,d,\mu_w)$ supports a $(1,1)$-Poincaré inequality, though the differentiability result does not rely on the Poincaré property to be proved. Building on Schioppa's approach to mutually singular doubling measures, this result extends to Laakso spaces and highlights a rich interplay between differentiability, measure theory, and PI-geometry in non-Euclidean settings. The framework sets the stage for analogous constructions in other self-similar Cantor-type spaces and clarifies how measure-theoretic tools can yield universal differentiability phenomena beyond the Euclidean paradigm.

Abstract

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincaré inequality.

Universal Differentiability Sets in Laakso Space

TL;DR

This work proves that Laakso space admits a continuous family of mutually singular doubling measures for which every Lipschitz function is differentiable almost everywhere with respect to each , thereby producing measure-zero universal differentiability sets in . The authors construct on the Cantor component and push forward to obtain , with the corresponding sets of full -measure whose complements have Hausdorff measure zero and which are mutually singular for distinct . A key feature is that each supports a -Poincaré inequality, though the differentiability result does not rely on the Poincaré property to be proved. Building on Schioppa's approach to mutually singular doubling measures, this result extends to Laakso spaces and highlights a rich interplay between differentiability, measure theory, and PI-geometry in non-Euclidean settings. The framework sets the stage for analogous constructions in other self-similar Cantor-type spaces and clarifies how measure-theoretic tools can yield universal differentiability phenomena beyond the Euclidean paradigm.

Abstract

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincaré inequality.
Paper Structure (11 sections, 18 theorems, 70 equations, 2 figures)

This paper contains 11 sections, 18 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular Doubling Measures on $K$ and $F$
  4. Measures on $K$
  5. Measures on $F$
  6. Rademacher's Theorem for a Singular Measure
  7. Measure Theoretic Preliminaries
  8. Auxilliary Sets
  9. Choice of Suitable Line Segments
  10. Estimate Along Line Segments
  11. Poincaré inequality

Key Result

Theorem 1.1

There exist doubling measures $\mu_{w}$ on $F$ for each $w\in (0,1)$ so that

Figures (2)

  • Figure 1: The path $\gamma_s$. The dashed line shows where the wormhole is used.
  • Figure 2: The three cases for rectangles that we consider. One of the sets is shaded gray and the other set is shaded black. In the middle case, the gray shading overlaps with the black shading. The dashed lines show wormhole levels. Notice how Case C can be decomposed into parts, which are related via Case A and Case B.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 3.1
  • ...and 26 more