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Finer geometry of planar self-affine sets

Balázs Bárány, Antti Käenmäki, Han Yu

Abstract

For planar self-affine sets satisfying the strong separation condition, recent work of Bárány, Hochman, and Rapaport gives mild assumptions under which the Hausdorff dimension equals the affinity dimension. In this paper, we study dominated systems in that regime and ask which finer geometric properties can be characterized. In the range $\dim_{\mathrm{H}}(X) < 1$, we characterize Ahlfors regularity by equivalent conditions involving positivity of $\mathcal{H}^s(X)$, control of projection fibers, and the identity $\dim_{\mathrm{L}}(X)=\dim_{\mathrm{H}}(X)=\dim_{\mathrm{A}}(X)$. In the range $\dim_{\mathrm{H}}(X) \ge 1$, we identify the maximal slice dimension as $\dim_{\mathrm{A}}(X)-1$ in Furstenberg directions and provide examples showing that Marstrand-type all-slice bounds cannot hold in general. We also derive projection consequences for Assouad dimension and exhibit dominated irreducible examples with $\dim_{\mathrm{aff}}(X)<\dim_{\mathrm{A}}(X)$.

Finer geometry of planar self-affine sets

Abstract

For planar self-affine sets satisfying the strong separation condition, recent work of Bárány, Hochman, and Rapaport gives mild assumptions under which the Hausdorff dimension equals the affinity dimension. In this paper, we study dominated systems in that regime and ask which finer geometric properties can be characterized. In the range , we characterize Ahlfors regularity by equivalent conditions involving positivity of , control of projection fibers, and the identity . In the range , we identify the maximal slice dimension as in Furstenberg directions and provide examples showing that Marstrand-type all-slice bounds cannot hold in general. We also derive projection consequences for Assouad dimension and exhibit dominated irreducible examples with .

Paper Structure

This paper contains 17 sections, 47 theorems, 224 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Main contributions
  4. Open problems
  5. Preliminaries
  6. Ahlfors regularity
  7. Weak tangents
  8. Irreducibility
  9. Domination
  10. Equilibrium states
  11. Self-affine set
  12. Main results
  13. Lower dimension of self-affine sets
  14. Assouad dimension of self-affine sets having large Hausdorff dimension
  15. Assouad dimension of self-affine sets having small Hausdorff dimension
  16. ...and 2 more sections

Key Result

Corollary 1.1

If $X$ is a dominated irreducible planar self-affine set satisfying the strong separation condition, then $X$ is Ahlfors $s$-regular if and only if $0 \leqslant s \leqslant 1$ and $\mathcal{H}^s(X)>0$.

Figures (2)

  • Figure 1: Schematic construction used in the proof of Corollary \ref{['thm:affinity-assouad']}. The first-level pieces are arranged so that projected overlaps are uniformly excluded in Furstenberg directions, which yields projective separation and then Ahlfors regularity via Theorem \ref{['thm:ahl']}.
  • Figure 2: Illustration of $H(x,V,\delta,\varepsilon)$ as the union of two opposite cones around the line $V+x$, separated from $x$ by the offset $\varepsilon$. As $\delta \downarrow 0$, these cones collapse to the punctured line $(V+x)\setminus B(x,\varepsilon)$.

Theorems & Definitions (86)

  • Corollary 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 76 more