Self-affine measures and vector-valued representations

• Autorzy: Qi-Rong Deng , Xing-Gang He , Ka-Sing Lau
• Języki publikacji: EN

Abstrakty

EN

Let A be a d × d integral expanding matrix and let $S_{j}(x) = A^{-1}(x + d_{j})$ for some $d_{j} ∈ ℤ^{d}$, j = 1,...,m. The iterated function system (IFS) ${S_{j}}_{j=1}^{m}$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof is constructive; it depends on using a tiling IFS ${ψ_{j}}_{j=1}^{l}$ to obtain a graph directed system, together with the associated probability on the vertices to form some transition matrices. As applications, we study the dimension and Lebesgue measure of a self-affine set, the $L^{q}$-spectrum of a self-similar measure, and the existence of a scaling function (i.e., an L¹-solution of the refinement equation).

Kategorie tematyczne

• 52C20: Tilings in 2 dimensions
• 52C22: Tilings in n dimensions
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2008
• Tom: 188
• Numer: 3
• Strony: 259-289

Daty

wydano

2008

Twórcy

autor

Qi-Rong Deng

• Department of Mathematics, Fujian Normal University, Fuzhou 350007, China
• Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

autor

Xing-Gang He

• Department of Mathematics, Central China Normal University, Wuhan 430079, China

autor

Ka-Sing Lau

• Department of Mathematics, The Chinese University of Hong Kong, Hong Kong

Identyfikatory

DOI

10.4064/sm188-3-3