Multidimensional self-affine sets: non-empty interior and the set of uniqueness

TL;DR

This paper investigates the conditions under which self-affine sets have non-empty interior and analyzes the size and structure of points with unique addresses, revealing new insights into their geometric and measure-theoretic properties.

Contribution

It establishes a threshold for the matrix determinant ensuring non-empty interior and characterizes the set of points with unique addresses, including a full description for special cases.

Findings

If | ext{det} M| extgreater= 2^{-1/d}, the attractor has non-empty interior.

The Hausdorff dimension of the set of points with unique addresses is positive for most matrices.

A complete description of the unique address set is provided for a special class of matrices.

Abstract

Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set $\mathcal U_M$ of points in $A_M$ which have a unique address. We show that unless $M$ belongs to a very special (non-generic) class, the Hausdorff dimension of $\mathcal U_M$ is positive. For this special class the full description of $\mathcal U_M$ is given as well. This paper continues our work begun in two previous papers.