Related papers: Dimensions of random affine code tree fractals
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 investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair'. In particular, we compute these dimensions for certain classes of self-affine sets and quasi-self-similar sets…
The fractal or Hausdorff dimension is a measure of roughness (or smoothness) for time series and spatial data. The graph of a smooth, differentiable surface indexed in $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface…
We study a model of random $\mathcal{R}$-enriched trees that is based on weights on the $\mathcal{R}$-structures and allows for a unified treatment of a large family of random discrete structures. We establish distributional limits…
We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring…
In this paper we study two random analogues of the box-like self-affine attractors introduced by Fraser, itself an extension of Sierpi\'nski carpets. We determine the almost sure box-counting dimension for the homogeneous random case…
We show a new method of estimating the Hausdorff measure (of the proper dimension) of a fractal set from below. The method requires computing the subsequent closest return times of a point to itself.
We study the fractal properties of the stationary distrubtion {\pi} for a simple Markov process on R. We will give bounds for the Hausdorff dimension of {\pi}, and lower bounds for the multifractal spectrum of {\pi}. Additionally, we will…
We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive iterated…
We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate…
We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model…
This paper presents a simple method of calculating the Hausdorff dimension for a class of non-conformal fractals.
We calculate the spectral dimension of a wide class of tree-like fractals by solving the random walk problem through a new analytical technique, based on invariance under generalized cutting-decimation transformations. These fractals are…
Let $W$ be a finite Weyl group and ${\hat{W}}$ be the corresponding affine Weyl group. We show that a large element in ${\hat{W}}$, randomly generated by (reduced) multiplication by simple generators, almost surely has one of $|W|$-specific…
We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension.
We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically…
We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from…
A simple method of calculating the Hausdorff-Besicovitch dimension of the Kronecker Product based fractals is presented together with a compact R script realizing it. The proposed new formula is based on traditionally used values of the…
We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realization of a fractal…