Related papers: $L^q$-spectra of self-affine measures: closed form…
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…
We study the generalized q-dimensions of measures supported on non-autonomous conformal attractors, which are the generalizations of Moran sets and the attractors of iterated function systems. We first prove that the critical values of…
Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this…
We exhibit an asymmetry phenomenon for uniqueness sets in $\ell^q$. Specifically, we construct sets that do not support measures with $\ell^q$-summable Fourier coefficients, yet simultaneously support measures whose positive frequencies…
Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin…
Spectral decomposition of matrices is a recurring and important task in applied mathematics, physics and engineering. Many application problems require the consideration of matrices of size three with spectral decomposition over the real…
We establish a connection between the $L^{q}$-spectrum of a Borel measure $\nu $ on the $m$-dimensional unit cube and the approximation order of Kolmogorov diameters of the unit sphere with respect to Sobolev norms in $L_{\nu }^{p}$. This…
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine…
We introduce two new concepts, local homogeneity and local L^q-spectrum, both of which are tools that can be used in studying the local structure of measures. The main emphasis is given to the examination of local dimensions of measures in…
An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension…
Let \mu_{M,D} be the self-similar measure generated by the positive integer M=RN^q and the product-form digit set D=\{0,1,\dots,N-1\}\oplus N^{p_1}\{0,1,\dots,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\dots,N-1\}, where R>1, N>1, q, p_i(1\leq…
In this article, we investigate the bivariate multifractal analysis of pairs of Borel probability measures. We prove that, contrarily to what happens in the univariate case, the natural extension of the Legendre spectrum does not yield an…
We study Falconer's subadditive pressure function with emphasis on analyticity. We begin by deriving a simple closed form expression for the pressure in the case of diagonal matrices and, by identifying phase transitions with zeros of…
We prove preservation of $L^q$ dimensions (for $1<q\le 2$) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures…
Multifractal formalism is designed to describe the distribution at small scales of the elements of $\mathcal M^+_c(\R^d)$, the set of positive, finite and compactly supported Borel measures on $\R^d$. It is valid for such a measure $\mu$…
We study spectral measures generated by infinite convolution products of discrete measures generated by Hadamard triples, and we present sufficient conditions for the measures to be spectral, generalizing a criterion by Strichartz. We then…
We have accurately measured the bispectrum for four scale-free models of structure formation with the spectral index $n=1$, 0, -1, and -2. The measurement is based on a new method that can effectively eliminate the alias and numerical…
The $L^q$ dimensions, for $1<q<\infty$, quantify the degree of smoothness of a measure. We study the following problem on the real line: when does the $L^q$ dimension improve under convolution? This can be seen as a variant of the…
In this note we study the $L^p-L^q$ boundedness of Fourier multipliers of anharmonic oscillators, and as a consequence also of spectral multipliers, for the range $1<p \leq 2 \leq q <\infty$. The underlying Fourier analysis is associated…
We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of…