# Hausdorff Dimension of Planar Self-Affine Sets and Measures with Overlaps

## TL;DR

This paper establishes that for certain planar self-affine measures with overlaps, the Hausdorff dimension equals the Lyapunov dimension under specific irreducibility and separation conditions, extending previous results to overlapping systems.

## Contribution

It extends the understanding of the dimension theory of self-affine measures to include cases with overlaps, under irreducibility and separation assumptions.

## Key findings

- Dimension equals Lyapunov dimension under specified conditions
- Results apply to totally irreducible systems and some reducible systems
- Advances the theory of self-affine measures with overlaps

## Abstract

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also treat a class of reducible systems. This extends our previous work on the subject with B\'ar\'any to the overlapping case.