On normal numbers and self-similar measures

Self-Similar Measures

Self-Similar Measures for Quasicrystals

We study self-similar measures of Hutchinson type, defined by compact families of contractions, both in a single and multi-component setting. The results are applied in the context of general model sets to infer, via a generalized version of Weyl’s Theorem on uniform distribution, the existence of invariant measures for families of self-similarities of regular model sets.

Cauchy Transforms of Self-Similar Measures

is a useful tool for numerical studies of the measure, since the measure of any reasonable set may be obtained as the line integral of F around the boundary. We give an effective algorithm for computing F when is a self-similar measure, based on a Laurent expansion of F for large z and a transformation law (Theorem 2.2) for F that encodes the self-similarity of . Using this algorithm we compute...

Convolutions of Equicontractive Self-similar Measures on the Line

Let μ be a self-similar measure on R generated by an equicontractive iterated function system. We prove that the Hausdorff dimension of μ∗n tends to 1 as n tends to infinity, where μ∗n denotes the n-fold convolution of μ. Similar results hold for the Lq dimension and the entropy dimension of μ∗n.

Laplace operators related to self - similar measures on R d

Given a bounded open subset Ω of Rd (d 1) and a positive finite Borel measure μ supported on Ω with μ(Ω) > 0, we study a Laplace-type operator μ that extends the classical Laplacian. We show that the properties of this operator depend on the multifractal structure of the measure, especially on its lower L∞-dimension dim∞(μ). We give a sufficient condition for which the Sobolev space H 1 0 (Ω) i...