Two-dimensional self-affine sets with interior points, and the set of uniqueness. (1st December 2015)

• Record Type: Journal Article
• Authors: Hare, Kevin G; Sidorov, Nikita
• Is Part Of: Nonlinearity. Volume 29:Number 1(2016:Jan.)
• Page Start: 1
• Page End: 26
• Publication Date: 2015-12-01
• Subjects: iterated function system -- self-affine set -- set of uniqueness; Primary 28A80 -- Secondary 11A67; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/1/1
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Let M be a real matrix with both eigenvalues less than 1 in modulus. Consider two self-affine contraction maps from, where . 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 . 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 [011 ].