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Point-wise doubling indices of measures and its application to bi-Lipschitz classification of Bedford-McMullen carpets

Hui Rao, Yan-Li Xu, Yuan Zhang

TL;DR

The paper introduces a point-wise doubling index $\delta_\varphi$ for non-doubling measures and applies it to the uniform Bernoulli measures on Bedford-McMullen carpets, proving these indices are Lipschitz invariants. By deriving explicit formulas for upper and lower indices using a reverse run-length function $\beta(k;\boldsymbol{\omega})$ and a secondary index $\gamma$, the authors establish new computable invariants that constrain bi-Lipschitz classifications. They show that, away from a small exceptional class, bi-Lipschitz equivalence forces carpets to share the same fiber sequence up to permutation and that vertical separation conditions are preserved, linking these indices to fiber geometry and multifractal spectra. Collectively, the work provides a robust framework for distinguishing BM-carpets beyond classical dimensions, enabling sharper Lipschitz classification results and offering deeper insight into the interaction between measure-theoretic growth and geometric structure.

Abstract

Doubling measure was introduced by Beurling and Ahlfors in 1956 and now it becomes a basic concept in analysis on metric space. In this paper, for a measure which is not doubling, we introduce a notion of point-wise doubling index, and calculate the point-wise doubling indices of uniform Bernoulli measures on Bedford-McMullen carpets. As an application, we show that, except a small class of Bedford-McMullen carpets, if two Bedford-McMullen carpets are bi-Lipschitz equivalent, then they have the same fiber sequence up to a permutation.

Point-wise doubling indices of measures and its application to bi-Lipschitz classification of Bedford-McMullen carpets

TL;DR

The paper introduces a point-wise doubling index for non-doubling measures and applies it to the uniform Bernoulli measures on Bedford-McMullen carpets, proving these indices are Lipschitz invariants. By deriving explicit formulas for upper and lower indices using a reverse run-length function and a secondary index , the authors establish new computable invariants that constrain bi-Lipschitz classifications. They show that, away from a small exceptional class, bi-Lipschitz equivalence forces carpets to share the same fiber sequence up to permutation and that vertical separation conditions are preserved, linking these indices to fiber geometry and multifractal spectra. Collectively, the work provides a robust framework for distinguishing BM-carpets beyond classical dimensions, enabling sharper Lipschitz classification results and offering deeper insight into the interaction between measure-theoretic growth and geometric structure.

Abstract

Doubling measure was introduced by Beurling and Ahlfors in 1956 and now it becomes a basic concept in analysis on metric space. In this paper, for a measure which is not doubling, we introduce a notion of point-wise doubling index, and calculate the point-wise doubling indices of uniform Bernoulli measures on Bedford-McMullen carpets. As an application, we show that, except a small class of Bedford-McMullen carpets, if two Bedford-McMullen carpets are bi-Lipschitz equivalent, then they have the same fiber sequence up to a permutation.
Paper Structure (10 sections, 20 theorems, 96 equations, 1 figure)

This paper contains 10 sections, 20 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Estimates of measures of approximate squares
  3. Lemmas
  4. Reverse run length function
  5. Proofs of Theorem \ref{['thm:lower']} and Theorem \ref{['thm:formula']}
  6. Remarks on $\delta_{-\log r}$ and $\delta_{\log|\log r|}$
  7. Run length
  8. Remarks on $\delta$ and $\Delta$
  9. A point-wise doubling index $\gamma$
  10. Proofs of Theorems \ref{['thm_5']}, \ref{['thm:fiber']} and \ref{['thm:best']}

Key Result

Lemma 1.1

The uniform Bernoulli measure $\mu_E$ is not doubling if and only if $a_0a_{m-1}>0$, $a_0\neq a_{m-1}$, and $a_ja_{j+1}> 0$ for at least one $j\in \{0,1,\dots,m-2\}$.

Figures (1)

  • Figure 1:

Theorems & Definitions (47)

  • Definition 1.1
  • Remark 1.1
  • Lemma 1.1
  • Definition 1.2: Miao2013 YangZh23
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Example 1.1
  • Theorem 1.3
  • ...and 37 more