arXiv subjects

Evgeny Verbitskiy

Publications and source records attributed to Evgeny Verbitskiy.

At least 19 recordsLinked to original sources

arXiv2025

The Generalized Friendship Paradox for Spectral Centralities

We revisit the classical friendship paradox which states that on an average ones friends have at least as many friends as oneself and generalize it to a variety of network centrality indices. For a broad class of spectral centralities on connected undirected graphs degree, eigenvector centrality, walk counts, Katz centrality and PageRank, we show that the average centrality of a nodes neighbours always exceeds the global average centrality.

cs.SI↗

arXiv2025

Gibbs Properties of Equilibrium States

We consider the problem of equivalence of Gibbs states and equilibrium states for continuous potentials on full shift spaces $E^{\mathbb{Z}}$. Sinai, Bowen, Ruelle and others established equivalence under various assumptions on the potential $ϕ$. At the same time, it is known that every ergodic measure is an equilibrium state for some continuous potential. This means that the equivalence can occur only under some appropriate conditions on the potential function. In this paper, we identify the necessary and sufficient conditions for the equivalence.

math.DS↗

arXiv2025

Computation of Lyapunov exponents of matrix products

For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(ω_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists, $$L(ω):=\lim_{n\to \infty} \frac 1n \log \|A_{ω_0} A_{ω_2}\cdots A_{ω_{n-1}}\|$$ defines the Lyapunov exponent of the sequence of matrices $(A_{ω_n})_{n\ge 0}$. It is proved that the Lyapunov exponent $L(ω)$ has a closed-form expression under certain conditions. One special case arises when $A_j$'s are non-negative and $ω$ is generic with respect to some shift-invariant measure; a second special case occurs when $A_j$'s (for $1\le j<m$) are invertible and $ω$ is a typical point with respect to some shift-ergodic measure. Substitutive sequences and characteristic sequences of $\mathcal{B}$-free integers are considered as examples. An application is presented for the computation of multifractal spectrum of weighted Birkhoff averages.

math.DS↗

arXiv2024

On an extension of a theorem by Ruelle to long-range potentials

Ruelle's transfer operator plays an important role in understanding thermodynamic and probabilistic properties of dynamical systems. In this work, we develop a method of finding eigenfunctions of transfer operators based on comparing Gibbs measures on the half-line $\mathbb Z_+$ and the whole line $\Z$. For a rather broad class of potentials, including both the ferromagnetic and antiferromagnetic long-range Dyson potentials, we are able to establish the existence of integrable, but not necessarily continuous, eigenfunctions. For a subset thereof we prove that the eigenfunction is actually continuous.

math.DS↗

arXiv2024

Bohr chaoticity of principal algebraic actions and Riesz product measures

For a continuous $\mathbb{N}^d$ or $\mathbb{Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb{Z}$ actions of positive entropy are Bohr-chaotic. The same is proved for principal algebraic $\mathbb{Z}^d$ ($d\ge 2$) actions of positive entropy under the condition of existence of summable homoclinic points.

math.DS↗

arXiv2024

Multifractal Formalism from Large Deviations

It has often been observed that the Multifractal Formalism and the Large Deviation Principles are intimately related. In fact, Multifractal Formalism was heuristically derived using the Large Deviations ideas. In numerous examples in which the multifractal results have been rigorously established, the corresponding Large Deviation results are valid as well. Moreover, the proofs of multifractal and large deviations are remarkably similar. The natural question then is whether under which conditions multifractal formalism can be deduced from the corresponding large deviations results. More specifically, given a sequence of random variables $\{ {X_n} \}_{n\in\N}$, satisfying a Large Deviation Principle, what can be said about the multifractal nature of the level sets $K_α=\{ω: \lim_{n} \frac{X_n(ω)}{n}=α\}$. Under some technical assumptions, we establish the upper and lower bounds for multifractal spectra in terms of the large deviation rate functions, and show that many known results of multifractal formalism are covered by our setup.

math.DS↗

arXiv2023

Pseudo-random number generation with $β$-encoders

The $β$-encoder is an analog circuit that converts an input signal $x \in [0,1]$ into a finite bit stream $\{b_i\}$. The bits $\{b_i\}$ are correlated and therefore are not immediately suitable for random number generation, but they can be used to generate bits $\{a_i\}$ that are (nearly) uniformly distributed. In this article we study two such methods. In the first part the bits $\{a_i\}$ are defined as the digits of the base-2 representation of the original input $x$. Under the assumption that there is no noise in the amplifier we then study a question posed by Jitsumatsu and Matsumura on how many bits $b_1, \ldots, b_m$ are needed to correctly determine the first $n$ bits $a_1,\ldots,a_n$. In the second part we show this method fails for random amplification factors. Nevertheless, even in this case, nearly uniformly distributed bits can still be generated from $b_1,\ldots,b_m$ using modern cryptographic techniques.

math.DS↗

arXiv2021

Random Lochs' Theorem

In 1964 Lochs proved a theorem on the number of continued fraction digits of a real number $x$ that can be determined from just knowing its first $n$ decimal digits. In 2001 this result was generalised to a dynamical systems setting by Dajani and Fieldsteel, where it compares sizes of cylinder sets for different transformations. In this article we prove a version of Lochs' Theorem for random dynamical systems as well as a corresponding Central Limit Theorem. The main ingredient for the proof is an estimate on the asymptotic size of the cylinder sets of the random system in terms of the fiber entropy. To compute this entropy we provide a random version of Rokhlin's formula for entropy.

math.DS↗

arXiv2021

Invariant densities for random continued fractions

We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.

math.DS↗

arXiv2021

Critical intermittency in random interval maps

Critical intermittency stands for a type of intermittent dynamics in iterated function systems, caused by an interplay of a superstable fixed point and a repelling fixed point. We consider critical intermittency for iterated function systems of interval maps and demonstrate the existence of a phase transition when varying probabilities, where the absolutely continuous stationary measure changes between finite and infinite. We discuss further properties of this stationary measure and show that its density is not in $L^q$ for any $q > 1$. This provides a theory of critical intermittency alongside the theory for the well studied Manneville-Pomeau maps, where the intermittency is caused by a neutral fixed point.

math.DS↗

arXiv2021

Decimation limits of principal algebraic $\mathbb{Z}^d$-actions

Let $f$ be a Laurent polynomial in $d$ commuting variables with integer coefficients. Associated to $f$ is the principal algebraic $\mathbb{Z}^d$-action $\alpha_f$ on a compact subgroup $X_f$ of $\mathbb{T}^{\mathbb{Z}^d}$ determined by $f$. Let $N\ge1$ and restrict points in $X_f$ to coordinates in $N\mathbb{Z}^d$. The resulting algebraic $N\mathbb{Z}^d$-action is again principal, and is associated to a polynomial $g_N$ whose support grows with $N$ and whose coefficients grow exponentially with $N$. We prove that by suitably renormalizing these decimations we can identify a limiting behavior given by a continuous concave function on the Newton polytope of $f$, and show that this decimation limit is the negative of the Legendre dual of the Ronkin function of $f$. In certain cases with two variables, the decimation limit coincides with the surface tension of random surfaces related to dimer models, but the statistical physics methods used to prove this are quite different and depend on special properties of the polynomial.

math.DS↗

arXiv2021

On regularity of functions of Markov chains

We consider processes which are functions of finite-state Markov chains. It is well known that such processes are rarely Markov. However, such processes are often regular in the following sense: the distant past values of the process have diminishing influence on the distribution of the present value. In the present paper, we present novel sufficient conditions for regularity of functions of Markov chains.

math.PR↗

arXiv2016

A Wiener Lemma for the discrete Heisenberg group: Invertibility criteria and applications to algebraic dynamics

This article contains a Wiener Lemma for the convolution algebra $\ell^1(\mathbb H,\mathbb C)$ and group $C^\ast$-algebra $C^\ast(\mathbb H)$ of the discrete Heisenberg group $\mathbb H$. At first, a short review of Wiener's Lemma in its classical form and general results about invertibility in group algebras of nilpotent groups will be presented. The known literature on this topic suggests that invertibility investigations in the group algebras of $\mathbb H$ rely on the complete knowledge of $\widehat{\mathbb H}$ -- the dual of $\mathbb H$, i.e., the space of unitary equivalence classes of irreducible unitary representations. We will describe the dual of ${\mathbb H}$ explicitly and discuss its structure. Wiener's Lemma provides a convenient condition to verify invertibility in $\ell^1(\mathbb H,\mathbb C)$ and $C^\ast(\mathbb H)$ which bypasses $\widehat{\mathbb H}$. The proof of Wiener's Lemma for $\mathbb H$ relies on local principles and can be generalised to countable nilpotent groups. As our analysis shows, the main representation theoretical objects to study invertibility in group algebras of nilpotent groups are the corresponding primitive ideal spaces. Wiener's Lemma for $\mathbb H$ has interesting applications in algebraic dynamics and Time-Frequency Analysis which will be presented in this article as well.

math.DS↗

arXiv2015

Thermodynamics of the Binary Symmetric Channel

We study a hidden Markov process which is the result of a transmission of the binary symmetric Markov source over the memoryless binary symmetric channel. This process has been studied extensively in Information Theory and is often used as a benchmark case for the so-called denoising algorithms. Exploiting the link between this process and the 1D Random Field Ising Model (RFIM), we are able to identify the Gibbs potential of the resulting Hidden Markov process. Moreover, we obtain a stronger bound on the memory decay rate. We conclude with a discussion on implications of our results for the development of denoising algorithms.

math.DS↗

arXiv2015

The random continued fraction transformation

We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the Rényi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.

math.DS↗

arXiv2014

Algebraic actions of the discrete Heisenberg group: Expansiveness and homoclinic points

We survey some of the known criteria for expansiveness of principal algebraic actions of countably infinite discrete groups. In the special case of the discrete Heisenberg group we propose a new approach to this problem based on Allan's local principle. Furthermore, we present a first example of an absolutely summable homoclinic point for a nonexpansive action of the discrete Heisenberg group and use it to construct an equal-entropy symbolic cover of the system.

math.DS↗

arXiv2012

Homoclinic points, atoral polynomials, and periodic points of algebraic Z^d-actions

Cyclic algebraic Z^d-actions are defined by ideals of Laurent polynomials in d commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative d-torus. For such expansive actions it is known that the limit for the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the d-torus in a finite set. Here we further extend it to the case when the dimension of intersection of the variety with the d-torus is at most d-2. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

math.DS↗

arXiv2010

Erasure entropies and Gibbs measures

Recently Verdu and Weissman introduced erasure entropies, which are meant to measure the information carried by one or more symbols given all of the remaining symbols in the realization of the random process or field. A natural relation to Gibbs measures has also been observed. In his short note we study this relation further, review a few earlier contributions from statistical mechanics, and provide the formula for the erasure entropy of a Gibbs measure in terms of the corresponding potentia. For some 2-dimensonal Ising models, for which Verdu and Weissman suggested a numerical procedure, we show how to obtain an exact formula for the erasure entropy. l

math-ph↗