Two-dimensional self-affine sets with interior points, and the set of uniqueness

TL;DR

This paper investigates the geometric properties of self-affine sets generated by specific contraction maps, establishing conditions for interior points and analyzing the structure of points with unique addresses.

Contribution

It provides new conditions under which self-affine attractors have interior points and characterizes the set of points with unique addresses for a broad class of IFS.

Findings

Self-affine sets have interior points when eigenvalues are between 0.8409 and 1.

The set of points with a unique address has positive Hausdorff dimension in most cases.

Exceptional cases with different properties are fully characterized.

Abstract

Let $M$ be a $2\times2$ real matrix with both eigenvalues less than~1 in modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to \mathbb R^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*} where $u\neq0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. Our two main results are as follows: 1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. 2. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension -- with the exceptional cases fully described as well. This paper continues our work begun in [11].