Related papers: Hausdorff Dimension of Average Conformal Hyperboli…

We propose a new algorithm for generating pseudorandom (pseudo-generic) numbers of conformal measures of a continuous map T acting on a compact space X and for a Holder continuous potential F. In particular, we show that this algorithm…

We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in this setting. This is a function field analogue of the…

In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow…

In this paper we study the singular set in the parabolic obstacle problem for general obstacles $\varphi \in C^{2,1}$. We prove that the singular set has parabolic Hausdorff dimension at most $n-1$. Prior to our result, this was only known…

Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff…

We explicitly calculate the Hausdorff dimension of the graph and range of an isotropic stable L\'{e}vy process $X$ plus deterministic drift function $f$. For that purpose we use a restricted version of the genuine Hausdorff dimension which…

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…

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…

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation u_t+\beta(x)u-\Delta u&=f(x,u),&&(t,x)\in[0,+\infty[\times\Omega,…

Given a compact basic semi-algebraic set we provide a numerical scheme to approximate as closely as desired, any finite number of moments of the Hausdorff measure on the boundary of this set. This also allows one to approximate interesting…

Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try…

Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian…

Let $\theta$ be an irrational number and $\varphi: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_\varphi(\theta) =\Big\{y \in \mathbb R: \|n\theta- y\|<\varphi(n), \ {\text{for infinitely…

We prove that every hyperbolic measure invariant under a C^{1+\alpha} diffeomorphism of a smooth Riemannian manifold possesses asymptotically ``almost'' local product structure, i.e., its density can be approximated by the product of the…

For the simple random walks in percolation clusters on hyperbolic groups, we show that the associated harmonic measures are exact dimensional and their Hausdorff dimensions are equal to the entropy over the speed. Our method is inspired by…

Specification is an important concept in dynamical systems introduced by Bowen. Schmeling proved that the set of $\beta>1$ such that the corresponding $\beta$-shift has specification is of Hausdorff dimension $1$. Hu et al. proved that the…

Z. Wen and J. Wu introduced the notion of homogeneous perfect sets as a generalization of Cantor type sets and determined their exact Hausdorff dimension based on the length of their fundamental intervals and the gaps between them. In this…

Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree T. Abert and Virag showed that there exist finitely generated (but…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

The aim of this paper is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this paper, three equivalent definitions for decomposition complexity are established. We…