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Local dimension spectrum for dominated planar self-affine sets

Alex Batsis, Antti Käenmäki, Tom Kempton

TL;DR

The paper analyzes the local dimension spectrum for dominated planar self-affine measures by building a thermodynamic formalism on the shift space, including a detailed treatment of singular-value potentials and the pressure $P(\psi^{q,s})$. It derives differentiability of the pressure, links the symbolic spectrum $\tau(q)$ to the Lyapunov cross-dimension via equilibrium states, and establishes a Legendre transform relation with the symbolic multifractal spectrum. Under projective strong separation and related conditions, it proves exact dimensionality for projections and obtains multifractal results for level sets $X_s$, including upper bounds and, in key regimes, full multifractal formalism. Additionally, it provides a construction (with random translations) showing absolute continuity of projected measures in a broad class, highlighting the role of projection properties in determining dimension spectra. Together, these results extend local dimension analyses from almost-sure to deterministic settings in 2D dominated self-affine systems and advance the understanding of multifractal structure in self-affine geometry.

Abstract

The local dimension spectrum provides a framework for quantifying the fractal properties of a measure, and it is well understood for non-overlapping self-similar measures. In this article, we study the local dimension spectrum for dominated self-affine measures. By analyzing exact dimensionality, we obtain deterministic results that extend the scope of the local dimension spectrum beyond the almost-sure setting.

Local dimension spectrum for dominated planar self-affine sets

TL;DR

The paper analyzes the local dimension spectrum for dominated planar self-affine measures by building a thermodynamic formalism on the shift space, including a detailed treatment of singular-value potentials and the pressure . It derives differentiability of the pressure, links the symbolic spectrum to the Lyapunov cross-dimension via equilibrium states, and establishes a Legendre transform relation with the symbolic multifractal spectrum. Under projective strong separation and related conditions, it proves exact dimensionality for projections and obtains multifractal results for level sets , including upper bounds and, in key regimes, full multifractal formalism. Additionally, it provides a construction (with random translations) showing absolute continuity of projected measures in a broad class, highlighting the role of projection properties in determining dimension spectra. Together, these results extend local dimension analyses from almost-sure to deterministic settings in 2D dominated self-affine systems and advance the understanding of multifractal structure in self-affine geometry.

Abstract

The local dimension spectrum provides a framework for quantifying the fractal properties of a measure, and it is well understood for non-overlapping self-similar measures. In this article, we study the local dimension spectrum for dominated self-affine measures. By analyzing exact dimensionality, we obtain deterministic results that extend the scope of the local dimension spectrum beyond the almost-sure setting.
Paper Structure (10 sections, 22 theorems, 147 equations)

This paper contains 10 sections, 22 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Thermodynamic Formalism
  3. Shift Space
  4. Lyapunov Dimension
  5. Domination
  6. Equilibrium States
  7. Differentiability of the Pressure
  8. Symbolic Spectrum
  9. Exact Dimensionality
  10. Multifractal Formalism

Key Result

Theorem 1.1

Suppose that $(A_1,\ldots, A_N) \in GL_2(\mathbb R)^N$ is such that $\|A_i\|<\frac{1}{2}$ for all $i \in \{1,\ldots,N\}$ and $(p_1,\ldots,p_N)$ is a probability vector. Given a choice of translation vector $\mathbf{v}=(v_1,\ldots,v_N)$, let $\mu_{\mathbf{v}}$ be the corresponding self-affine measure for Lebesgue almost every choice of $\mathbf{v}$.

Theorems & Definitions (42)

  • Theorem 1.1: Barral-Feng BarralFeng
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 32 more