Semicontinuity of dimension and measure for locally scaling fractals
The basic question of this paper is: If you consider two iterated function systems close to each other in an appropriate topology, are the dimensions of their respective invariant sets close to each other? It is well known that the Hausdorff dimension (and Lebesgue measure) of the invariant set does not depend continuously on the iterated function system. Our main result is that (with a restriction on the “non-conformality” of the transformations) the Hausdorff dimension is a lower semicontinuous function in the $C^1$-topology of the transformations of the iterated function system. The same question is raised of the Lebesgue measure of the invariant set. Here we show that it is an upper semicontinuous function of the transformations. We also include some corollaries of these results, such as the equality of box and Hausdorff dimensions in these cases.