SELF-SIMILAR MEASURES AND THEIR FOURIER TRANSFORMS-III

A self-similar family of measures on R(n) is defined to be a family of finite positive Borel measures {mu(u)}u is-an-element-of V indexed by the vertices V of a connected directed multigraph which satisfies [GRAPHICS] where E(uv), is the set of edges joining u to v, S(e) are contractive similarities and p(e) are positive weights satisfying [GRAPHICS] Similarly, we define a self-conformal family by allowing S(e) to be conformal maps. We compute pointwise and L(p) dimensions for such measures, provided the mappings S(e) satisfy appropriate separation hypotheses. The pointwise dimension is defined by [GRAPHICS] if the limit exists, and the L(p) dimension for 1 < p < infinity is defined by [GRAPHICS] if the limit exists. We also consider self-similar families of distributions of compact support which satisfy (*) with fewer conditions on the weights p(e), and we give an existence theorem based on the Fourier transform.