Assouad dimension of planar self-affine sets

TL;DR

This paper computes the Assouad dimension of certain planar self-affine sets, demonstrating their minimality for conformal Assouad dimension and revealing strong tangential regularity properties.

Contribution

It provides explicit calculation of Assouad dimension for self-affine sets under specific conditions and establishes their tangential regularity and minimality properties.

Findings

Assouad dimension of the sets is explicitly calculated.

Sets are minimal for conformal Assouad dimension.

Any two generic points share the same tangent sets.

Abstract

We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set $X$ adheres to very strong tangential regularity by showing that any two points of $X$, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.