Fourier decay for self-similar measures

Large Scale Renormalisation of Fourier Transforms of Self-similar Measures and Self-similarity of Riesz Measures

We shall show that the oscillations observed by Strichartz JRS92, Str90] in the Fourier transforms of self-similar measures have a large-scale renormali-sation given by a Riesz-measure. Vice versa the Riesz measure itself will be shown to be self-similar around every triadic point.

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.

Self-Similar Measures

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...

Multifractal formalism for self-similar measures with weak separation condition

For any self-similar measure μ on R satisfying the weak separation condition, we show that there exists an open ball U0 with μ(U0) > 0 such that the distribution of μ, restricted on U0, is controlled by the products of a family of non-negative matrices, and hence μ|U0 satisfies a kind of quasi-product property. Furthermore, the multifractal formalism for μ|U0 is valid on the whole range of dime...