On a family of self-affine sets: Topology, uniqueness, simultaneous expansions

Abstract

Let β 1 , β 2 > 1 and T i (x, y) = x+i β1 , y+i β2 , i ∈ {±1}. Let A := A β1,β2 be the unique compact set satisfying A = T 1 (A) ∪ T −1 (A). In this paper we give a detailed analysis of A, and the parameters (β 1 , β 2 ) where A satisfies various topological properties. In particular, we show that if β 1 < β 2 < 1.202, then A has a nonempty interior, thus significantly improving the bound from [2]. In the opposite direction, we prove that the connectedness locus for this family studied in [15] is not simply connected. We prove that the set of points of A which have a unique address has positive Hausdorff dimension for all (β 1 , β 2 ). Finally, we investigate simultaneous (β 1 , β 2 )-expansions of reals, which were the initial motivation for studying this family in [5].

Digit frequencies and self-affine sets with non-empty interior

“…One problem the authors of these papers were particularly interested in was determining those pairs (β 1 , β 2 ) for which the attractor Λ β 1 ,β 2 has non-empty interior. The best result in this direction is the following result due to Hare and Sidorov [22].…”

“…When β 2 = β 3 we denote Λ β 1 ,β 2 ,β 3 by Λ β 1 ,β 2 . The case where β 2 = β 3 was studied in [12] and [22]. One problem the authors of these papers were particularly interested in was determining those pairs (β 1 , β 2 ) for which the attractor Λ β 1 ,β 2 has non-empty interior.…”

“…Before moving onto our proofs we say a few words about the methods used in this paper and compare them with those used in [12] and [22]. In these papers the authors show that (0, 0) ∈ Λ o β 1 ,β 2 by constructing a polynomial P (x) = x n +b n−1 x n−1 +b 1 x+b 0 which satisfies:…”

Iterated Functions Systems, Blenders, and Parablenders

“…In [HS1,HS2], its is proved that the IFS generated by ( f +1 , f −1 ) has non empty interior. Let us adapt their proof to obtain the following stronger result.…”

“…Proof. From [HS1,Theorem 3.4], there exists a monic polynomial Q(x) = x n + a n−1 x n−1 + · · · + a 0 such that n−1 i=0 |a i | < 2 and (x − 1) N |Q(x). Dividing by some x k , one can assumes that a 0 = 0.…”