Hausdorff measure of uniform self-similar fractals
نویسنده
چکیده
Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on Rd. Denote by D the Hausdorff dimension, by HD(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on ω, such that the Hausdorff dimension is smaller than one and HD(E) = diam(E)D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where ω depends on c and HD(E) < diam(E)D, if c is small enough. MSC(2010): 28A80, 28A78 keywords: Self-similar set, Hausdorff measure
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