On the dimension of self-affine sets and measures with overlaps

TL;DR

This paper computes the Hausdorff dimension of self-affine sets and measures with overlaps in the plane, using advanced techniques to handle overlaps and provide bounds on exceptional parameters.

Contribution

It extends the understanding of self-affine sets with overlaps by combining Hochman and Feng, Hu's techniques to compute dimensions and bound exceptional sets.

Findings

Computed Hausdorff dimension of self-affine attractors and measures.

Provided upper bounds for the dimension of exceptional parameter sets.

Extended techniques for analyzing overlaps in self-affine systems.

Abstract

In this paper we consider diagonally affine, planar IFS $\Phi=\left\{S_i(x,y)=(\alpha_ix+t_{i,1},\beta_iy+t_{i,2})\right\}_{i=1}^m$. Combining the techniques of Hochman and Feng, Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.