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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Self-affine measures and vector-valued representations

### Tom 188 / 2008

Studia Mathematica 188 (2008), 259-289 MSC: Primary 52C20, 52C22; Secondary 28A80. DOI: 10.4064/sm188-3-3

#### Streszczenie

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).

#### Autorzy

• Qi-Rong DengDepartment of Mathematics
Fujian Normal University
Fuzhou 350007, China
and
Department of Mathematics
The Chinese University of Hong Kong
Hong Kong
e-mail
• Xing-Gang HeDepartment of Mathematics
Central China Normal University
Wuhan 430079, China
e-mail
• Ka-Sing LauDepartment of Mathematics
The Chinese University of Hong Kong
Hong Kong
e-mail

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