The Hausdorff dimension of self-projective sets

TL;DR

This paper investigates the Hausdorff dimension of attractors generated by projective actions of finite sets of matrices in SL(2,R), extending recent results and exploring dimension continuity and measure support.

Contribution

It generalizes prior work by establishing the Hausdorff dimension as the minimum of 1 and the critical exponent under specific conditions, using combined techniques from IFS and Möbius semigroup theories.

Findings

Dimension equals min(1, critical exponent) under certain conditions

Continuity of Hausdorff dimension established

Dimension of Furstenberg measure support discussed

Abstract

Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a recent result of Solomyak and Takahashi by showing that the Hausdorff dimension of $K_\mathcal{A}$ is given by the minimum of 1 and the critical exponent, under the assumption that $\mathcal{A}$ satisfies certain discreteness conditions and a Diophantine property. Our approach combines techniques from the theories of iterated function systems and M\"obius semigroups, and allows us to discuss the continuity of the Hausdorff dimension, as well as the dimension of the support of the Furstenberg measure.