相关论文: The dimension of projections of fractal percolatio…
In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion…
We consider fractal percolation (or Mandelbrot percolation) which is one of the most well studied example of random Cantor sets. Rams and the first author studied the projections (orthogonal, radial and co-radial) of fractal percolation…
A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost…
In this paper we consider fractal percolation random Cantor sets $E$ on the plane constructed with non-homogeneous probabilities. We focus on the case when the probabilities are large enough to guarantee that the almost sure dimension of…
We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less…
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…
This is a survey paper about the fractal percolation process, also known as Mandelbrot percolation. It is intended to give a general breadth overview of more recent research in the topic, but also includes some of the more classical…
We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of $\mathbb{R}^2$…
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…
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…
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…
We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire…
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.
Let F1 and F2 be independent copies of correlated fractal percolation, with Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will contain an…
In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost…
We prove that, with probability 1, all orthogonal projections of the natural measure on a percolation fractal are absolutely continuous and (except for the horizontal and vertical projection) have H\"older continuous density.
Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the…
The interstellar medium seems to have an underlying fractal structure which can be characterized through its fractal dimension. However, interstellar clouds are observed as projected two-dimensional images, and the projection of a…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study…