Related papers: Quantitative recurrence problem on some Bedford-Mc…
In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem,…
The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a…
We give a complete solution to the Mandelbrot-Kahane problem for the microcanonical cascade measures by determing their exact Fourier dimensions. We also discuss the Frostman regularity as well as the bi-H\"older continuity of the…
We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales H\"older continuous) regularity of such…
We prove that any non-degenerate Bedford-McMullen carpet does not admit oblique self-embedding similitudes; that is, if $f$ is a similitude sending the carpet into itself, then the image of the $x$-axis under $f$ must be parallel to one of…
In this paper we generalize in Lorentz-Minkowski space $\l^3$ the two-dimensional analogue of the catenary of Euclidean space. We solve the Dirichlet problem for bounded mean convex domains and spacelike boundary data that have a spacelike…
In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference…
We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all…
Period doubling H\'enon renormalization of strongly dissipative maps is generalized in arbitrary finite dimension. In particular, a small perturbation of toy model maps with dominated splitting has invariant $C^r$ surfaces embedded in…
In 1996 Y. Kifer obtained a variational formula for the Hausdorff dimension of the set of points for which the frequencies of the digits in the Cantor series expansion is given. In this note we present a slightly different approach to this…
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…
In this note we discuss three interconnected problems about dynamics of Hamiltonian or, more generally, just smooth diffeomorphisms. The first two concern the existence and properties of the maps whose iterations approximate the identity…
We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic…
Let $A$ be a $2\times 2$ integral matrix with an eigenvalue of modulus strictly less than 1. Let $T$ be the natural endomorphism on the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, induced by $A$. Given $\tau>0$, let \[ R_\tau =\{\, x\in…
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs…
For every tuple $d_1,\dots, d_l\geq 2,$ let $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}$ denote the tensor product of $\mathbb{R}^{d_i},$ $i=1,\dots,l.$ Let us denote by $\mathcal{B}(d)$ the hyperspace of centrally symmetric…
The canonical dimension is an invariant attached to admissible representations of p-adic reductive groups, which has only received significant attention in the case of mod-p representations. In the case of complex representations, the…
Let $f$ be an endomorphism of $\mathbb{CP}^k$ and $\nu$ be an $f$-invariant measure with positive Lyapunov exponents $(\lambda_1,\...,\lambda_k)$. We prove a lower bound for the pointwise dimension of $\nu$ in terms of the degree of $f$,…
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…
In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz…