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Exceptional projections of self-affine sets: an introduction

Ian D. Morris

TL;DR

This paper addresses the problem of determining the Hausdorff dimension of projections of self-affine fractals and proves a sharp upper bound for projected attractors in terms of a projected affinity dimension $\text{dim}_{\mathsf{aff}}^Q \mathsf{A}$. The authors provide a self-contained Falconer-style proof of the upper bound and introduce the projected pressure $P_Q(\mathsf{A},s)$ to analyze when projections are exceptional, i.e., when $\text{dim}_{\mathsf{H}} QX < \text{dim}_{\mathsf{aff}} \mathsf{A}$. They then construct explicit, highly structured examples (via Kronecker products and split orthogonal groups) where large families of projections are exceptional, including cases where isotropic planes yield a dimension drop with orientation-dependent behavior. These constructions show that exceptional projection sets can be rich and are not limited to simple subspace kernels, providing new insight into Marstrand-type theorems for self-affine sets. Overall, the work extends the dimension theory for self-affine sets to projected images, clarifies the role of projection-induced dimension drops, and supplies concrete methods to realize and study manifolds within exceptional projection sets.

Abstract

We describe some recent results on the dimensions of linear projections of self-affine fractals, focusing in particular on an upper bound for the dimension of the projected image. We give a self-contained treatment of this bound and illustrate it through explicit examples, in the process exhibiting some smooth submanifolds of the Grassmannian which can be contained in the exceptional set in Marstrand's theorem.

Exceptional projections of self-affine sets: an introduction

TL;DR

This paper addresses the problem of determining the Hausdorff dimension of projections of self-affine fractals and proves a sharp upper bound for projected attractors in terms of a projected affinity dimension . The authors provide a self-contained Falconer-style proof of the upper bound and introduce the projected pressure to analyze when projections are exceptional, i.e., when . They then construct explicit, highly structured examples (via Kronecker products and split orthogonal groups) where large families of projections are exceptional, including cases where isotropic planes yield a dimension drop with orientation-dependent behavior. These constructions show that exceptional projection sets can be rich and are not limited to simple subspace kernels, providing new insight into Marstrand-type theorems for self-affine sets. Overall, the work extends the dimension theory for self-affine sets to projected images, clarifies the role of projection-induced dimension drops, and supplies concrete methods to realize and study manifolds within exceptional projection sets.

Abstract

We describe some recent results on the dimensions of linear projections of self-affine fractals, focusing in particular on an upper bound for the dimension of the projected image. We give a self-contained treatment of this bound and illustrate it through explicit examples, in the process exhibiting some smooth submanifolds of the Grassmannian which can be contained in the exceptional set in Marstrand's theorem.
Paper Structure (11 sections, 19 theorems, 66 equations, 4 figures)

This paper contains 11 sections, 19 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction and background
  2. Hausdorff dimension and iterated function systems
  3. Falconer's theorem on self-affine sets
  4. Projections of fractal sets and an extension of Falconer's theorem
  5. Proof of the upper dimension bound
  6. Construction of examples
  7. Underlying principles and preliminaries
  8. A concrete example constructed via Kronecker products
  9. A general example constructed via Kronecker products
  10. Examples arising from the split orthogonal group
  11. Acknowledgements

Key Result

Theorem 1

Let $T_1,\ldots,T_N \colon \mathbb{R}^d \to \mathbb{R}^d$ be contractions of $\mathbb{R}^d$. Then there exists a unique nonempty compact set $X\subset \mathbb{R}^d$ such that $X=\bigcup_{i=1}^N X$. If additionally every $T_i$ is a similarity transformation which contracts the Euclidean norm by preci

Figures (4)

  • Figure 1: Self-similar sets satisfying the hypotheses of Theorem \ref{['th:hutch']} may be efficiently covered using sets of the form $T_\mathtt{i} \mathbf{B}$ where $\mathbf{B}$ is a ball which contains the attractor.
  • Figure 2: For self-affine sets which are not self-similar, naı̈ve application of the covering strategy described §\ref{['ss:onepointone']} and illustrated in Figure \ref{['fi:covers']} in general results in an inefficient cover using long, thin ellipsoids. To derive an efficient cover one dissects these ellipsoids into a larger number of smaller but rounder shapes.
  • Figure 3: Two projections of the attractor of a system $(T_1, T_2)$ whose linearisation is the example $\mathsf{A}$ defined in Proposition \ref{['pr:old']} and whose additive parts are given by $v_1:=0$ and $v_2:=(1,0,1,0)^T$. The first projection corresponds to a linear map of the form $Q \otimes I$ and the second to a map of the form $I\otimes Q$; the latter results in a substantial apparent dimension drop.
  • Figure 4: Two projections of the attractor of the system $(T_1, T_2)$ defined in \ref{['eq:so22-example']} onto isotropic planes of differing orientations.

Theorems & Definitions (32)

  • Theorem 1
  • Proposition 1.1: Singular value decomposition
  • Proposition 1.2: Singular value identities and inequalities
  • Theorem 2: Falconer
  • Theorem 3: Marstrand's theorem
  • Theorem 4: Feng-Xie FeXi25, Morris-Sert MoSe25
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • ...and 22 more