Finer geometry of planar self-affine sets

TL;DR

This paper investigates the detailed geometric properties of planar self-affine sets under certain conditions, revealing new characterizations of regularity, slice dimensions, and projection behaviors, especially in relation to Hausdorff and affinity dimensions.

Contribution

It provides new characterizations of geometric regularity and slice dimensions for self-affine sets, and explores the relationship between various fractal dimensions in this context.

Findings

Ahlfors regularity characterized by positivity of Hausdorff measure and dimension equalities.

Maximal slice dimension identified as affinity dimension minus one in certain directions.

Counterexamples showing limitations of Marstrand-type bounds and differences between affinity and Assouad dimensions.

Abstract

For planar self-affine sets satisfying the strong separation condition, recent work of B\'ar\'any, 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 $\dim_{\mathrm{aff}}(X)<\dim_{\mathrm{A}}(X)$.