Self-similar sets in complete metric spaces

Abstract

We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that K K has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.