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

Abstract

Let M be a real matrix with both eigenvalues less than 1 in modulus. Consider two self-affine contraction maps from , Tm(v)=Mv−u  and  Tp(v)=Mv+u, where . We are interested in the properties of the attractor of the iterated function system (IFS) generated by Tm and Tp, i.e. the unique non-empty compact set A such that . Our two main results are as follows: • If both eigenvalues of M are between and 1 in absolute value, and the IFS is non-degenerate, then A has non-empty interior. • 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 [].