Rigidity Properties of Locally Scaling Fractals

A set has local scaling if in a neighborhood of a point the structure of the set can be mapped onto a finer scale structure of the set. These scaling transformations are compact sets of locally affine contractions (that is, contractions with uniformly α-Hölder continuous derivatives). In this setting, without the open set condition or any other assumption on the spacing of these contractions, we show that the measure of the set is an upper semi-continuous function of the scaling transformations in the C-topology. With a restriction on the ’non-conformality’ of the scaling transformations we show that the Hausdorff dimension is a lower semi-continuous function in the C-topology. We include some examples to show that continuity does not follow in either case. 1