Digit sets of integral self-affine tiles with prime determinant

• Autorzy: Jian-Lin Li
• Języki publikacji: EN

Abstrakty

EN

Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ${ϕ_{d}(x) = M^{-1}(x+d)}_{d∈ D}$ has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that $D = M^{γ}D₀$, where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results of Kenyon, Lagarias and Wang. We then give several remarks and examples to illustrate some problems on digit sets.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 177
• Numer: 2
• Strony: 183-194

Daty

wydano

2006

Twórcy

autor

Jian-Lin Li

• College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, P.R. China

Identyfikatory

DOI

10.4064/sm177-2-7