The dimension of projections of self-affine sets and measures

TL;DR

This paper investigates the projection dimensions of self-affine sets and measures, establishing that under certain conditions, most projections retain the full Hausdorff dimension except possibly in one direction.

Contribution

It proves that self-affine measures with equal Hausdorff and Lyapunov dimensions have projections of maximal dimension in all but one direction, extending to self-affine sets.

Findings

Projections of certain self-affine measures have dimension min{dim_H mu, 1} in almost all directions.

Many self-affine sets have projections of dimension min{dim_H E, 1} in all but at most one direction.

The results connect measure-theoretic and geometric properties of self-affine fractals.

Abstract

Let E be a plane self-affine set defined by affine transformations with linear parts given by matrices with positive entries. We show that if mu is a Bernoulli measure on E with dim_H mu = dim_L mu, where dim_H and dim_L denote Hausdorff and Lyapunov dimensions, then the projection of mu in all but at most one direction has Hausdorff dimension min{dim_H mu,1}. We transfer this result to sets and show that many self-affine sets have projections of dimension min{dim_H E,1} in all but at most one direction.