Self-affine measures and vector-valued representations
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AbstractLet A be a d x d integral expanding matrix and let S(j)(x) = A(-1)(x + d(j)) for some d(j) is an element of 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 {psi(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 L(1)-solution of the refinement equation).
All Author(s) ListDeng QR, He XG, Lau KS
Journal nameStudia Mathematica
Volume Number188
Issue Number3
Pages259 - 286
LanguagesEnglish-United Kingdom
Keywordsdigit set; Hausdorff dimension; L(q)-spectrum; scaling function; self-affine; tile; vector-valued measure
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-16-01 at 00:08