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The dimension of projections of planar diagonal self-affine measures

Aleksi Pyörälä

TL;DR

The paper proves that planar diagonal self-affine measures $\mu$, defined by diagonal IFS with an irrationality condition on the diagonal entries, have full projection dimension in every non-principal direction: for all $\theta$, $\dim_{\rm H} \pi_{\theta}\mu = \min\{1, \dim_{\rm H}\mu\}$. The authors employ the local entropy averages framework of Hochman–Shmerkin, developing a filtration that yields approximate squares and, crucially, a product structure for magnifications of the measure. They treat separately the cases $\lambda_1^{\mu} > \lambda_2^{\mu}$ (where vertical flattening leads to a clean product form) and $\lambda_1^{\mu} = \lambda_2^{\mu}$ (handled via Robbins’ equidistribution to reduce to the previous case). A key technical tool is a uniform lower bound on small-scale entropies of projected measures, established via a detailed product-projection proposition and its local-variant analysis, which, combined with ergodicity and Marstrand-type projection results, yields the claimed projection dimension formula. This advances projection theory for self-affine objects under weaker separation assumptions and highlights the role of irrationality and Lyapunov structure in controlling exceptional sets of directions.

Abstract

We show that if $μ$ is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, $$ \dim_{\rm H} μ\circ π^{-1}= \min\lbrace 1,\dim_{\rm H}μ\rbrace $$ for any non-principal orthogonal projection $π$.

The dimension of projections of planar diagonal self-affine measures

TL;DR

The paper proves that planar diagonal self-affine measures , defined by diagonal IFS with an irrationality condition on the diagonal entries, have full projection dimension in every non-principal direction: for all , . The authors employ the local entropy averages framework of Hochman–Shmerkin, developing a filtration that yields approximate squares and, crucially, a product structure for magnifications of the measure. They treat separately the cases (where vertical flattening leads to a clean product form) and (handled via Robbins’ equidistribution to reduce to the previous case). A key technical tool is a uniform lower bound on small-scale entropies of projected measures, established via a detailed product-projection proposition and its local-variant analysis, which, combined with ergodicity and Marstrand-type projection results, yields the claimed projection dimension formula. This advances projection theory for self-affine objects under weaker separation assumptions and highlights the role of irrationality and Lyapunov structure in controlling exceptional sets of directions.

Abstract

We show that if is a self-affine measure on the plane defined by an iterated function system of contractions with diagonal linear parts, then under an irrationality assumption on the entries of the linear parts, for any non-principal orthogonal projection .
Paper Structure (13 sections, 13 theorems, 101 equations, 3 figures, 1 table)

This paper contains 13 sections, 13 theorems, 101 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. On the proof
  3. Preliminaries
  4. Entropy
  5. Self-affine measures
  6. Suspension flows
  7. Local entropy averages
  8. The filtration
  9. Proof of Theorem \ref{['theorem-main']}
  10. The case $\lambda_1^\mu > \lambda_2^\mu$
  11. Product structure
  12. The case $\lambda_1^\mu= \lambda_2^\mu$
  13. Proof of Proposition \ref{['prop-productprojections']}

Key Result

Theorem 1.1

Let $\Phi=\lbrace \varphi_i(x) = A_i x + a_i\rbrace_{i\in\Gamma}$ be a diagonal affine iterated function system on $\mathbb R^2$. Suppose that Then for any self-affine measure $\mu$ associated to $\Phi$ and for any $\theta\in\mathbb R$,

Figures (3)

  • Figure 1: The rectangle $\Pi(E_n(\mathtt{i}))$ (colored gray) in the case a) on the right and in the case b) on the left.
  • Figure 2: Under the assumption $\lambda_1^\mu>\lambda_2^\mu$, the cylinders $\Pi([\mathtt{i}|_n])$ are "vertically flat". The "approximate square" $\Pi(E_n(\mathtt{i}))$ is colored gray.
  • Figure 3: The statement of Claim \ref{['productrepresentation']}. The measure $S_{-\log |\lambda_1(\mathtt{j}|_\ell)|}T_\ell\tilde{\Pi} (\bar{\mu} \times \bar{\mu}_\mathtt{i})_{F_\ell^\theta(\mathtt{j},\mathtt{k})}$ is a product of $\pi_\mathtt{x}\mu$ and $S_{\log |\lambda_2(\mathtt{k}|_{\tau_\ell})|-\log |\lambda_1(\mathtt{j}|_\ell)|}\mu_{\sigma^{\tau_\ell}\mathtt{k}} \approx S_{-\theta}\mu_{\sigma^{\tau_\ell}\mathtt{k}}$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1: Concavity
  • Lemma 2.2: Continuity
  • Lemma 2.3: Chain rule
  • Theorem 2.4: Corollary of Theorem 2.11 of FengHu2009
  • Proposition 2.5
  • proof
  • Remark 3.1
  • Theorem 3.2
  • Lemma 4.1
  • ...and 17 more