In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.
Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings ${f_{ij}}_{(i,j)in G}$ and let $mu$ be the self-affine measure associated with ${f_{ij}}_{(i,j)in G}$ and a probability vector $(p_{ij})_{(i,j)in G}$. We study the asymptotics of the geometric mean error in the quantization for $mu$. Let $s_0$ be the Hausdorff dimension for $mu$. Assuming a separation condition for ${f_{ij}}_{(i,j)in G}$, we prove that the $n$th geometric error for $mu$ is of the same order as $n^{-1/s_0}$.
We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Holder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates.
We refine the multifractal formalism for the local dimension of a Gibbs measure $mu$ supported on the attractor $Lambda$ of a conformal iterated functions system on the real line. Namely, for given $alphain mathbb{R}$, we establish the formalism for the Hausdorff dimension of level sets of points $xinLambda$ for which the $mu$-measure of a ball of radius $r_{n}$ centered at $x$ obeys a power law $r_{n}{}^{alpha}$, for a sequence $r_{n}rightarrow0$. This allows us to investigate the Holder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.
We introduce a probability distribution on $mathcal{P}([0,1]^d)$, the space of all Borel probability measures on $[0,1]^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero lower quasi-Assouad dimension (hence the upper and lower Assouad dimensions are almost surely infinite or zero). We also indicate how the results extend to other Assouad-like dimensions.
Given a compactly supported probability measure on a Riemannian manifold, we study the asymptotic speed at which it can be approximated (in Wasserstein distance of any exponent p) by finitely supported measure. This question has been studied under the names of ``quantization of distributions and, when p=1, ``location problem. When p=2, it is linked with Centroidal Voronoi Tessellations.