Related papers: On the Hausdorff dimension of regular points of in…

We determine the Hausdorff dimension of $k$-multiple points for a symmetric operator semistable L\'evy process $X=\{X(t), t\in\mathbb{R}_+\}$ in terms of the eigenvalues of its stability exponent. We also give a necessary and sufficient…

Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $\eta:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $\xi$, we denote by…

We show that the uniform Littlewood Conjecture (ULC) recently introduced by Bandi, Fregoli and Kleinbock is false. More precisely the counterexamples form a residual set, the method further suggests positive Hausdorff dimension. For a…

We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for…

We prove, with a more geometric approach, that the solutions to the Navier-Stokes equations are regular up to a set of Hausdorff dimension 1. The main tool for the proof is a new compactness lemma and the monotonicity property of harmonic…

We provide evidence that the small-scale statistics of the acceleration of a test particle in high-Reynolds number Lagrangian turbulence is correctly described by Tsallis statistics with entropic index q=3/2. We present theoretical…

The Kardar-Parisi-Zhang (KPZ) equation is a celebrated non-linear stochastic equation featuring non-equilibrium scaling. Although in one dimension, its statistical properties are very well understood, a new scaling regime has been reported…

The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\ge0}$ is contracting in the $L^1$-distance. For the…

The criterion of the recurrent compact set was introduced by Moreira and Yoccoz to prove that stable intersections of regular Cantor sets on the real line are dense in the region where the sum of their Hausdorff dimensions is bigger than 1.…

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…

Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \]…

We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two.

We study the evolution of a small perturbation of the equilibrium of a totally asymmetric one-dimensional interacting system. The model we take as example is Hammersley's process as seen from a tagged particle, which can be viewed as a…

The asymptotic behavior for fully coupled multiscale stochastic systems becomes much complicated when the fast processes do not locate in a compact space. An example is constructed to show that the averaged coefficients may become…

Let Q be an infinite set of positive integers. Denote by W_{\tau, n}(Q) (resp. W_{\tau, n}) the set of points in dimension n simultaneously \tau--approximable by infinitely many rationals with denominators in Q (resp. in N*). A non--trivial…

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…

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…

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the $L^1$ norm. As a…

We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow $v=(y,0)$ and a stratified temperature $\theta=\alpha y$ with $\alpha>\frac{1}{4}$. We show that for any $\epsilon>0$ there…

We compute the Hausdorff multifractal spectrum of two versions of multistable L{\'e}vy motions. These processes extend classical L{\'e}vy motion by letting the stability exponent $\alpha$ evolve in time. The spectra provide a decomposition…