Multidimensional Self-affine Sets: Non-empty Interior and the Set of Uniqueness

Let M be a d× d real contracting matrix. In this paper we consider the selfaffine iterated function system {Mv − u,Mv + u}, where u is a cyclic vector. Our main result is as follows: if | detM | ≥ 2−1/d, then the attractor AM has non-empty interior. We also consider the set UM of points in AM which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of UM is positive. For this special class the full description of UM is given as well. This paper continues our work begun in [5, 6]. 1. Non-empty interior Let d ≥ 2 and M be a d×d real matrix whose eigenvalues are all less than 1 in modulus. Denote by AM the attractor for the contracting self-affine iterated function system (IFS) {Mv − u,Mv + u}, i.e., AM = {πM(a0a1 . . . ) | an ∈ {±1}}, where πM(a0a1 . . . ) = ∞ ∑ k=0 akM u. If AM ∋ x = πM(a0a1 . . . ), then we call the sequence a0a1 · · · ∈ {±1}N an address of x. We assume our IFS to be non-degenerate, i.e., AM does not lie in any (d− 1)-dimensional subspace of R (i.e., AM spans R). Let u ∈ R be a cyclic vector for M , i.e., span{Mu | n ≥ 0} = R. Our main result is as follows. Theorem 1.1. If | detM | ≥ 2−1/d, then the attractor AM has non-empty interior. In particular, this is the case when each eigenvalue of M is greater than 2−1/d 2 in modulus. Remark 1.2. Note that if | detM | < 1 2 , then AM is a null set (see [4]) and therefore, has empty interior. It is an interesting question whether 2−1/d in Theorem 1.1 can be replaced with a constant independent of d. Date: July 6, 2015. 2010 Mathematics Subject Classification. 28A80.