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Projections of self-affine fractals

Ian Morris, Cagri Sert

TL;DR

The paper advances fractal geometry by proving a Falconer-type projection theorem for self-affine sets, yielding a stratified description of projected dimensions across algebraic subvarieties of Grassmannians. It introduces a projected thermodynamic formalism, defining a $Q$-projected affinity dimension $\mathrm{dim}_{\mathsf{aff}}^Q\mathsf{A}$ and $ (\varphi^s,Q)$-equilibrium states, and proves existence, Gibbs-type inequalities, and algebraicity of sublevel sets; these underpin almost-sure dimension formulas for projections and measures. The authors also construct robust examples showing new phenomena: (i) equilibrium measures on self-affine fractals with non-exact-dimensional projections, and (ii) strongly irreducible self-affine sets with small sumsets lacking arithmetic resonance. Together, these results extend Falconer’s framework to projections, connect to stratified Marstrand-type theorems, and reveal algebraic structures governing exceptional projections, with implications for the dimension theory of projected measures and fractal sumsets. The work integrates subadditive thermodynamic formalism, representation theory, and algebraic geometry to develop a comprehensive theory of projected dimensions, including the algebraicity of sublevel sets and stratified results that hold for Lebesgue almost every IFS under mild contractions and irreducibility assumptions.

Abstract

We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are preserved by the underlying linear algebraic group. The techniques which we develop allow us to construct examples of additional new phenomena: firstly, we give general examples of equilibrium measures on self-affine fractals which admit non-exact-dimensional projections. Secondly, we construct strongly irreducible self-affine sets which have small sumsets without any arithmetic resonance in their construction.

Projections of self-affine fractals

TL;DR

The paper advances fractal geometry by proving a Falconer-type projection theorem for self-affine sets, yielding a stratified description of projected dimensions across algebraic subvarieties of Grassmannians. It introduces a projected thermodynamic formalism, defining a -projected affinity dimension and -equilibrium states, and proves existence, Gibbs-type inequalities, and algebraicity of sublevel sets; these underpin almost-sure dimension formulas for projections and measures. The authors also construct robust examples showing new phenomena: (i) equilibrium measures on self-affine fractals with non-exact-dimensional projections, and (ii) strongly irreducible self-affine sets with small sumsets lacking arithmetic resonance. Together, these results extend Falconer’s framework to projections, connect to stratified Marstrand-type theorems, and reveal algebraic structures governing exceptional projections, with implications for the dimension theory of projected measures and fractal sumsets. The work integrates subadditive thermodynamic formalism, representation theory, and algebraic geometry to develop a comprehensive theory of projected dimensions, including the algebraicity of sublevel sets and stratified results that hold for Lebesgue almost every IFS under mild contractions and irreducibility assumptions.

Abstract

We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are preserved by the underlying linear algebraic group. The techniques which we develop allow us to construct examples of additional new phenomena: firstly, we give general examples of equilibrium measures on self-affine fractals which admit non-exact-dimensional projections. Secondly, we construct strongly irreducible self-affine sets which have small sumsets without any arithmetic resonance in their construction.

Paper Structure

This paper contains 33 sections, 21 theorems, 179 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context: the theorems of Falconer and Marstand
  3. A Falconer-type theorem for projections of self-affine sets
  4. Fractal sumsets
  5. Projections of measures
  6. Recent related work
  7. The setting and the main results
  8. Fundamental notations
  9. Thermodynamic formalism for linear images of self-affine sets
  10. Almost sure dimension formulas
  11. Examples
  12. Structure of the remainder of the article
  13. An abstract formulation of Theorem \ref{['th:pressure-detail']}
  14. Preliminaries from subadditive thermodynamic formalism
  15. Statement of the abstract theorem
  16. ...and 18 more sections

Key Result

Theorem A

Let $\mathsf{A}=(A_i)_{i \in \mathcal{I}} \in \mathop{\mathrm{GL}}\nolimits_d(\mathbb{R})^\mathcal{I}$ and suppose that $\max_{i \in \mathcal{I}} {\left\vert\left\vert\left\vert A_i \right\vert\right\vert\right\vert}<1$ for some norm ${\left\vert\left\vert\left\vert \cdot \right\vert\right\vert\righ

Figures (1)

  • Figure 1: Two projections of the attractor of an iterated function system whose linearisation generates a Zariski-dense subsemigroup of $\mathbb{R}^* \mathop{\mathrm{SO}}\nolimits(2,2)$. The two projections are onto two-dimensional isotropic subspaces of $\mathbb{R}^4$ with differing orientations.

Theorems & Definitions (36)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Theorem C
  • Theorem 1
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2
  • Corollary 2.3: Stratified Marstand-type projection theorem for self-affine sets
  • Remark 2.4
  • ...and 26 more