Related papers: Interpolating between Hausdorff and box dimension
We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U|…
The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function $h(\theta)$ to be realized as the intermediate…
The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta}\Lambda$, interpolates between its Hausdorff and box dimensions using the parameter $\theta\in[0,1]$. Determining a precise formula for…
Intermediate dimensions were recently introduced to provide a spectrum of dimensions interpolating between Hausdorff and box-counting dimensions for fractals where these differ. In particular, the self-affine Bedford-McMullen carpets are a…
We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $\theta…
We introduce a family of dimensions, which we call the $\Phi$-intermediate dimensions, that lie between the Hausdorff and box dimensions and generalise the intermediate dimensions introduced by Falconer, Fraser and Kempton. This is done by…
This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential theoretic methods are used to produce dimension bounds for images of sets under H\"older maps and…
Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain…
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…
$\theta$ intermediate dimensions are a continuous family of dimensions that interpolate between Hausdorff and Box dimensions of fractal sets. In this paper we study the problem of the relationship between the dimension of a set…
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,…
Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in…
In this paper, we define a family of dimensions for Borel measures that lie between the Hausdorff and Minkowski dimensions for measures, analogous to the intermediate dimensions of sets. Previously, Hare et. al. in [11] defined families of…
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff…
We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a Bara\'nski system and allow both vertical and horizontal random translations, while…
Intermediate dimensions are a class of new fractal dimensions which provide a spectrum of dimensions interpolating between the Hausdorff and box-counting dimensions. In this paper, we study the intermediate dimensions of Moran sets. Moran…
We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is…
Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is…
This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for…
We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are…