相关论文: Packing spectra for Bernoulli measures supported o…

In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford-McMullen. This extends the work of King where the multifractal analysis is computed…

We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $\mu_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling…

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…

Let $\theta$ be a Bernoulli measure which is stationary for a random walk generated by finitely many contracting rational affine dilations of $\mathbb{R}^d$, and let $\mathcal{K} = \mathrm{supp}(\theta)$ be the corresponding attractor. An…

We calculate the Assouad and lower dimensions of graph-directed Bedford-McMullen carpets, which reflect the extreme local scaling laws of the sets, in contrasting with known results on Hausdorff and box dimensions. We also investigate the…

Doubling measure was introduced by Beurling and Ahlfors in 1956 and now it becomes a basic concept in analysis on metric space. In this paper, for a measure which is not doubling, we introduce a notion of point-wise doubling index, and…

For a self-affine measure on a Bedford-McMullen carpet we prove that its quantization dimension exists and determine its exact value. Further, we give various sufficient conditions for the corresponding upper and lower quantization…

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of random self-affine Bedford-McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between…

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…

Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss…

We study Bedford--McMullen type carpets whose selected grid rectangles may be reflected in one or both coordinates. The organizing principle is that the Hausdorff dimension is controlled by the entropy of the weak-coordinate projection.…

In this paper we compute the Assouad and lower spectra of Bedford--McMullen sponges in $\mathbb{R}^3$ explicitly. According to the formulae established, we discover that the spectra are not determined by the ratio set, and the box, lower…

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum…

It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures $\mu$, both the…

Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1]^{\mathbb{N}}\times[0,1]^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of…

Besicovitch showed that if a set is null for the Hausdorff measure associated to a given dimension function, then it is still null for the Hausdorff measure corresponding to a smaller dimension function. We prove that this is not true for…

We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This…

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism $(x,y) \mapsto (mx \mod 1,ny \mod 1)$ that are Bernoulli measures for the natural Markov partition. We show that the statistics…

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…

We consider the multifractal analysis for Birkhoff averages of continuous potentials on a self-affine Sierpi\'{n}ski sponge. In particular, we give a variational principal for the packing dimension of the level sets. Furthermore, we prove…