Related papers: Hausdorff dimension of operator semistable L\'evy …
We compute the Hausdorff dimension of a two-dimensional Weierstrass function, related to lacunary (Hadamard gap) power series, that has no L\'evy area. This is done by interpreting it as a pullback attractor of a dynamical system based on…
An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.
In this paper we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles of Xiao (second author). As applications, uniform Hausdorff and…
Dilative semistability extends the notion of semi-selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. It is shown that this scaling relation is a natural extension…
For all $n \geq 2$, we construct a metric space $(X,d)$ and a quasisymmetric mapping $f\colon [0,1]^n \rightarrow X$ with the property that $f^{-1}$ is not absolutely continuous with respect to the Hausdorff $n$-measure on $X$. That is,…
In this note we connect Sobolev estimates in the context of polynomial averages e.g. \[ \| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m \] whenever some $f_i$ vanishes…
Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…
We prove that if $E\subseteq \R^2$ is analytic and $1<d < \dim_H(E)$, there are ``many'' points $x\in E$ such that the Hausdorff dimension of the pinned distance set $\Delta_x E$ is at least $d\left(1 -…
In this paper we study pseudo-processes related to odd-order heat-type equations composed with L\'evy stable subordinators. The aim of the article is twofold. We first show that the pseudo-density of the subordinated pseudo-process can be…
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…
In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods.…
We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider here, our methods require only $C^3$ regularity of the maps in…
For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of $\mathbb{R}^{2n}$ and for product sets.
Consider a random process s solution of the stochastic partial differential equation Ls = w with L a homogeneous operator and w a multidimensional L\'evy white noise. In this paper, we study the asymptotic effect of zooming in or zooming…
In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following…
Previous work has shown that the Hausdorff dimension of sofic affine-invariant sets is expressed as a limit involving intricate matrix products. This limit has typically been regarded as incalculable. However, in several highly non-trivial…
This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained…
Under weaker condition than that of Riedi & Mandelbrot, the Hausdorff (and Hausdorff-Besicovitch) dimension of infinite self-similar set K which is the invariant compact set of infinite contractive similarities {S_j(x)} satisfying open set…
If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2\lambda_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-\lambda_0\in(-1,-1/2)$ is the lead eigenvalue of a killed…
Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.