The Assouad dimension of self-affine carpets with no grid structure

TL;DR

This paper investigates the Assouad dimension of self-affine carpets lacking grid structure, revealing its connection to projections and local dimensions, and extending understanding beyond grid-based cases.

Contribution

It introduces a novel analysis of the Assouad dimension for non-grid self-affine carpets, linking it to projections and Bernoulli measures.

Findings

Assouad dimension relates to projections' box and Assouad dimensions.

The dimension is connected to local dimensions of Bernoulli measure projections.

Special case links Assouad dimension to Bernoulli convolutions.

Abstract

Previous study of the Assouad dimension of planar self-affine sets has relied heavily on the underlying IFS having a `grid structure', thus allowing for the use of approximate squares. We study the Assouad dimension of a class of self-affine carpets which do not have an associated grid structure. We find that the Assouad dimension is related to the box and Assouad dimensions of the (self-similar) projection of the self-affine set onto the first coordinate and to the local dimensions of the projection of a natural Bernoulli measure onto the first coordinate. In a special case we relate the Assouad dimension of the Przytycki-Urba\'nski sets to the lower local dimensions of Bernoulli convolutions.