Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set.We first extend the definition ofWSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V , the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V . © 2007 Elsevier Inc. All rights reserved