Self-affine measures and vector-valued representations
Tom 188 / 2008
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).
