Self-affine measures and vector-valued representations

Qi-Rong Deng¹ ; Xing-Gang He² ; Ka-Sing Lau³

¹Department of Mathematics Fujian Normal University Fuzhou 350007, China and Department of Mathematics The Chinese University of Hong Kong Hong Kong
²Department of Mathematics Central China Normal University Wuhan 430079, China
³Department of Mathematics The Chinese University of Hong Kong Hong Kong

Studia Mathematica, Tome 188 (2008) no. 3, pp. 259-289

Résumé

Let $A$ be a $d\times d$ integral expanding matrix and let $S_j(x) = A^{-1}(x+d_j)$ for some $d_j \in \mathbb{Z}^d$, $j =1, \dots, m$. 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 {\it open set condition}. In this paper we prove a general theorem on such representation. The proof is constructive; it depends on using a {\it 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).

Détail

DOI : 10.4064/sm188-3-3

Keywords: times integral expanding matrix mathbb dots iterated function system ifs generates self affine measures scale functions general ifs has overlaps known many special cases analysis measures functions facilitated expressing vector valued forms respect another ifs satisfies set condition paper prove general theorem representation proof constructive depends using tiling ifs psi obtain graph directed system together associated probability vertices form transition matrices applications study dimension lebesgue measure self affine set q spectrum self similar measure existence scaling function solution refinement equation