Related papers: Recurrence to shrinking targets on typical self-af…

Note by the author: Section 9.3 is added from the more general unpublished manuscript ``A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets'', (2021) by I. Kirat. Original abstract: An integral…

The diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly. The fractals considered are of the class generated iteratively by successive dilations and translations, and include generalizations of the…

In this paper, we investigate the Hausdorff measure of planar dominated self-affine sets at their affinity dimension. We show that the Hausdorff measure being positive and finite is equivalent to the K\"aenm\"aki measure being a mass…

We work with the Friedrichs extension of a one dimensional Schrodinger whose potential has a certain type of regular singularity near one end point. We study the effect on the eigenvalues of shrinking the region slightly near the end point.…

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their…

The behavior of the Hausdorff dimension of a set when projected onto a subspace is a fundamental question in fractal geometry. In this paper, we settle a question of Fassler and Orponen concerning the dimension of a set when projected onto…

We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions,…

In this work, we aim to advance the development of a fractal theory for sets of integers. The core idea is to utilize the fractal structure of $p$-adic integers, where $p$ is a prime number, and compare this with conventional densities and…

We consider limit sets of some conformal iterated function systems, and introduce classes of subsets of the limit set, with the property that the classes are closed under countable intersections and all sets in the classes have large…

Suppose $X$ is a compact connected metric space and $f: X \to X$ is a metric coarse expanding conformal map in the sense of Ha\"issinsky-Pilgrim. We show that if $X$ contains a homeomorphic copy of the letter "Y", then the Hausdorff…

We prove that if $\mu$ is a self-affine measure in the plane whose defining IFS acts totally irreducibly on $\mathbb{RP}^1$ and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension. We also…

There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…

For countably infinite IFSs on $\mathbb R^2$ consisting of affine contractions with diagonal linear parts, we give conditions under which the affinity dimension is an upper bound for the Hausdorff dimension and a lower bound for the lower…

We compute the almost sure Hausdorff dimension of random self-affine sponges in $\mathbb{R}^d$ without imposing any separation conditions. In this context, randomness arises from the matrices in the defining semigroup, which are random yet…

Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $\tau>0$, and a sequence $\{z_n\}$ of points in…

A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff…

We consider fixed-point equations for probability distributions on isometry classes of measured metric spaces. The construction is required to be recursive and tree-like, but we allow loops for the geodesics between points in the support of…

We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of…

An affine iterated function system is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self-affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension…