Hausdorff dimension of self-similar measures and sets with common fixed point structure

TL;DR

This paper investigates the Hausdorff dimension of self-similar measures and sets with shared fixed points, demonstrating weak exponential separation for most parameters and applying results to the generalized 4-corner set.

Contribution

It establishes weak exponential separation for systems with common fixed points, extending previous results, and computes the Hausdorff dimension of measures on the generalized 4-corner set.

Findings

Most systems satisfy weak exponential separation

Hausdorff dimension of self-affine measures on the 4-corner set determined

Exceptional parameter set has Hausdorff co-dimension one

Abstract

In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of a Hausdorff co-dimension one exceptional set of natural parameters, such systems satisfy a weak exponential separation. This significantly strengthens the previous result of the first author and Szv\'ak. As an application, we give the Hausdorff dimension of self-affine measures supported on the generalised 4-corner set.