BME Fractal Geometry Seminar

Multifractal analysis mainly focuses on describing the local behaviors of a function, a measure, or a stochastic process X on R^d. In this context, the pointwise behavior is measured by an exponent, which depends on the nature of the object. The multifractal spectrum D_X is then calculated, which describes the size of sets of points with the same regularity exponent.

The problem of prescribing the multifractal spectrum for a function or measure has been studied extensively by many researchers in multifractal analysis. However, a multivariate extension is essential because there are phenomena where the data are intrinsically composed of a family of correlated signals.

I will thus present two results extending the univariate construction of functions and 'homogeneous' measures with a prescribed multifractal spectrum to the bivariate case.

The \(\alpha\)-Weierstrass function is defined as \(W^{\alpha,b}_g(x) = \sum_{k=0}^{\infty} b^{-\alpha k} g(b^k x)\), where \(g\) is a Lipschitz function on the unit circle. For a prevalent \(\alpha\)-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most \(1-\alpha\), and the Hausdorff dimension of almost every level set equals \(1-\alpha\) with respect to its occupation measure. By this we answer a question raised by R. Anttila, B. Bárány, and A. Käenmäki. We further demonstrate that the occupation measure of a prevalent \(\alpha\)-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension \(d\), we construct Lipschitz functions \(g_0, g_1, \dots, g_{d-1}\) such that the mapping \(x \mapsto \big(W^{\alpha,b}_{g_0}(x), W^{\alpha,b}_{g_1}(x), \dots, W^{\alpha,b}_{g_{d-1}}(x)\big)\) is \(\alpha\)-bi-Hölder. We also prove that such an embedding requires at least \(1/\alpha\) coordinate functions.

This is a joint work with Antti Käenmäki and Balázs Maga.