Singularity of self-similar measures with respect to Hausdorff measures

Orthogonal Polynomials with Respect to Self-Similar Measures

We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theo...

Renewal of Singularity Sets of Statistically Self-similar Measures

This paper investigates new properties concerning the multifractal structure of a class of statistically self-similar measures. These measures include the well-known Mandelbrot multiplicative cascades, sometimes called independent random cascades. We evaluate the scale at which the multifractal structure of these measures becomes discernible. The value of this scale is obtained through what we ...

Convergence on Manifolds of Gibbs Measures Which Are Absolutely Continuous with Respect to Hausdorff Measures

We identify a sufficiant condition for a sequence of Gibbs measures Pλ with the density Zλe01, v ∈ Rn, defined on a state space of v′s, to converge weakly in a sense of measures to a Gibbs measure with a density Zλe −J0(v), where the dominating measure for the density is the Hausdorff measure with an appropriate dimension. The function J0 identifies an objective and J1 defines a constraint. The...

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.