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

作者：Hare Kevin G^*; Sidorov Nikita^*

来源：Nonlinearity, 2016, 29(1): 1-26.

DOI：10.1088/0951-7715/29/1/1

• iterated function system
• self-affine set
• set of uniqueness

Let M be a 2x2 real matrix with both eigenvalues less than 1 in modulus. Consider two self-affine contraction maps from R-2 -> R-2, T-m(v)= Mv-u and T-p(v)= Mv+ u,
where u not equal 0. 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) boolean OR T-p(A). Our two main results are as follows:
If both eigenvalues of M are between 2(-1/4) approximate to 0.8409 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 [11].

• 出版日期2016-1