Related papers: Multifractal level sets and metric mean dimension …

The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures under the open set condition the behavior of…

In this paper, we study the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai…

For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Let $s_n(x)=\sum_{j=1}^n a_j(x)$. The Hausdorff dimensions of the level sets $E_{\varphi(n),\alpha}:=\{x\in(0,1):…

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated…

We study the multifractal analysis of dimension spectrum for almost additive potential in a class of one dimensional non-uniformly hyperbolic dynamic systems and prove that the irregular set has full Hausdroff dimension.

We estimate the bounds of box-counting dimension of hidden variable fractal interpolation functions (HVFIFs) and hidden variable bivariate fractal interpolation functions (HVBFIFs) with four function contractivity factors and present…

This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show…

Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called multifractal spectrum. The practical…

We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend…

We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation…

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent…

Let $[a_1(x), a_2(x), \ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. We study the growth rate of the maximal product of consecutive partial quotients among the first $n$ terms, defined by…

The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…

A simple multifractal coarsening model is suggested that can explain the observed dynamical behavior of the fractal dimension in a wide range of coarsening fractal systems. It is assumed that the minority phase (an ensemble of droplets) at…

Let $M$ be smooth $n$-dimensional manifold, fibered over a $k$-dimensional submanifold $B$ as $\pi:M \to B$, and $\vartheta \in \Lambda^k (M)$; one can consider the functional on sections $\phi$ of the bundle $\pi$ defined by $\int_D \phi^*…

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet…

In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave…

Given $q\in\mathbb{N}$, let $(X,T)$ be a $\mathbb{Z}^q$-system, $\vec{v}\in\mathbb{R}^q\setminus\{\vec{0}\}$ be a direction vector and $\textbf{b}\in\mathbb{R}_+^{q-1}$. We study $(X,T)$ that has bounded complexity with respect to three…

Fluctuations in the return time statistics of a dynamical system can be described by a new spectrum of dimensions. Comparison with the usual multifractal analysis of measures is presented, and difference between the two corresponding sets…

Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_\mu(T)$, $\bar{h}_\mu(T)$ for any $\mu\in M(X)$, where $M(X)$ denotes the collection of all Borel probability measures…