# Open set condition and pseudo Hausdorff measure of self-affine IFSs

## TL;DR

This paper establishes a link between the open set condition and the positivity of pseudo Hausdorff measure for self-affine sets, extending classical results from self-similar to self-affine fractals.

## Contribution

It proves that the open set condition is equivalent to positive pseudo Hausdorff measure at the pseudo similarity dimension for self-affine IFSs, generalizing known self-similar results.

## Key findings

- Open set condition characterizes positive pseudo Hausdorff measure.
- Pseudo Hausdorff measure relates to an upper s-density with respect to a pseudo norm.
- The measure's exact value connects to a specific measure density in the self-affine context.

## Abstract

Let $A$ be an $n\times n$ real expanding matrix and $\mathcal{D}$ be a finite subset of $\mathbb{R}^n$ with $0\in\mathcal{D}$. The family of maps $\{f_d(x)=A^{-1}(x+d)\}_{d\in\mathcal{D}}$ is called a self-affine iterated function system (self-affine IFS). The self-affine set $K=K(A,\mathcal{D})$ is the unique compact set determined by $(A, {\mathcal D})$ satisfying the set-valued equation $K=\displaystyle\bigcup_{d\in\mathcal{D}}f_d(K)$. The number $s=n\,\ln(\# \mathcal{D})/\ln(q)$ with $q=|\det(A)|$, is the so-called pseudo similarity dimension of $K$. As shown by He and Lau, one can associate with $A$ and any number $s\ge 0$ a natural pseudo Hausdorff measure denoted by $\mathcal{H}_w^s.$ In this paper, we show that, if $s$ is chosen to be the pseudo similarity dimension of $K$, then the condition $\mathcal{H}_w^s(K)> 0$ holds if and only if the IFS $\{f_d\}_{d\in\mathcal{D}}$ satisfies the open set condition (OSC). This extends the well-known result for the self-similar case that the OSC is equivalent to $K$ having positive Hausdorff measure $\mathcal{H}^s$ for a suitable $s$. Furthermore, we relate the exact value of pseudo Hausdorff measure $\mathcal{H}_w^s(K)$ to a notion of upper $s$-density with respect to the pseudo norm $w(x)$ associated with $A$ for the measure $\mu=\lim\limits_{M\to\infty}\sum\limits_{d_0,\dotsc,d_{M-1}\in\mathcal{D}}\delta_{d_0 + Ad_1 + \dotsb + A^{M-1}d_{M-1}}$ in the case that $\#\mathcal{D}\le\lvert\det A\rvert$.