Assouad-type dimensions and the local geometry of fractal sets

Abstract

We study the fine local scaling properties of rough or irregular subsets of a metric space. In particular, we consider the classical Assouad dimension as well as two variants: a scale-refined variant called the Assouad spectrum, and a location-refined variant called the pointwise Assouad dimension.

For the Assouad spectrum, we first give a simple characterization of when a function 𝜑: (0,1) ⟶ [0,𝑑] can be the Assouad spectrum of a general subset of ℝᵈ. Using this, we construct examples exhibiting novel exotic behaviour, answering some questions of Fraser & Yu. We then compute the Assouad spectrum of a certain family planar self-affine sets: the class of Gatzouras–Lalley carpets. Within this family, we establish an explicit formula as the concave conjugate of a certain "column pressure" combined with simple parameter change. This class of sets exhibits novel behaviour in the setting of dynamically invariant sets, such as strict concavity and differentiability on the whole range (0,1).

We then focus on the interrelated concepts of (weak) tangents, Assouad dimension, and a new localized variant which we call the pointwise Assouad dimension. For general attractors of bi-Lipschitz iterated function systems, we establish that the Assouad dimension is given by the Hausdorff dimension of a tangent at some point in the attractor. Under the additional assumption of self-conformality, we moreover prove that this property holds for a subset of full Hausdorff dimension. We then turn our attention again to planar self-affine sets. For Gatzouras–Lalley carpets, we obtain precise information about tangents which, in particular, shows that points with large tangents are very abundant. However, already for Barański carpets, we see that more complex behaviour is possible.