## Institute for Mathematical Sciences

## Preprint ims97-2

** J.J.P. Veerman and L. Jonker**
* Rigidity Properties Of Locally Scaling Fractals*

Abstract: Local scaling of a set means that in a neighborhood of a point the structure of the set can be mapped into a finer scale
structure of the set. These scaling transformations are compact
sets of locally affine (that is: with uniformly
$\alpha$-H\"older continuous derivatives) contractions. In this
setting, without any assumption on the spacing of these
contractions such as the open set condition, we show that the
measure of the set is an upper semi-continuous of the scaling
transformation in the $C^0$-topology. With a restriction on the
'non-conformality' (see below) the Hausdorff dimension is
lower semi-continous function in the $C^{1}$-topology. We
include some examples to show that neither of these notions is
continuous.

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