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Marks Ruziboev

Publications and source records attributed to Marks Ruziboev.

17 recordsLinked to original sources
arXiv2026

On the dynamics of self-consistent transfer operators for interval maps with singularities

We study the dynamics of self-consistent transfer operators associated with mean-field coupled systems whose unit dynamics is given by an interval map with finitely many singularities of cusp type. We show that, when the coupling strength is sufficiently weak, such an operator has a unique fixed point in a suitable cone of the associated Banach space, and that this fixed point attracts the orbit of every density in the cone at an exponential rate in the corresponding norm.

math.DS↗
arXiv2026

Linear response for random systems with a cusp

We study i.i.d.\ random compositions of cusp tent-like interval maps having a common cusp point and a common cusp value. For cusp exponents $-1<\beta<-\frac12$, we prove that the associated annealed transfer operators have a spectral gap on $W^{1,1}(I)$ and $W^{2,1}(I)$. For all sufficiently small perturbations of the probability law, there is a stationary density $h_\varepsilon\in W^{2,1}(I)$, unique among stationary densities belonging to $W^{1,1}(I)$. Moreover, the map $\varepsilon\mapsto h_\varepsilon$ is differentiable at $\varepsilon=0$ in $W^{1,1}(I)$, and we obtain an explicit linear response formula.

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arXiv2024

Quenched Mixing Rates for Doubly Intermittent Maps

We study quenched mixing rates for random compositions of two classes of interval maps with two indifferent fixed points and a singularity at the origin: Pikovsky maps and Grossmann--Horner maps. For the Pikovsky family, each fibre map preserves Lebesgue measure, so the equivariant sample measures are given by \(\mu_\omega=m\). For the Grossmann--Horner family, we construct an equivariant family \((\mu_\omega)_{\omega\in\Omega}\) of absolutely continuous probability measures. Using random Young towers, we prove quenched polynomial decay of both future and past fibre correlations for bounded observables against H\"older observables. The rates are determined by quenched return time tail estimates obtained from endpoint drift bounds for the random cocycle.

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arXiv2023

Quenched decay of correlations for nonuniformly hyperbolic random maps with an ergodic driving system

In this article we study random tower maps driven by an ergodic automorphism. We prove quenched exponential correlations decay for tower maps admitting exponential tails. Our technique is based on constructing suitable cones of functions, defined on the random towers, which contract with respect to the Hilbert metric under the action of appropriate transfer operators. We apply our results to obtain quenched exponential correlations decay for several non-iid random dynamical systems including small random perturbations of Lorenz maps and Axiom A attractors.

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arXiv2023

On a linear differential game in the Hilbert space $\ell^2$

Two-player pursuit-evasion differential game and time optimal zero control problem in $\ell^2$ are considered. Optimal control for the corresponding zero control problem is found. A strategy for the pursuer that guarantees the solution for the pursuit problem is constructed.

math.OC↗
arXiv2021

Almost sure rates of mixing for partially hyperbolic attractors

We introduce random towers to study almost sure rates of correlation decay for random partially hyperbolic attractors. Using this framework, we obtain abstract results on almost sure exponential, stretched exponential and polynomial correlation decay rates. We then apply our results to small random perturbations of Axiom A attractors, small perturbations of derived from Anosov partially hyperbolic systems and to solenoidal attractors with random intermittency.

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arXiv2021

On the stability and null-controllability of an infinite system of linear differential equations

In this work, the null controllability problem for a linear system in $\ell^2$ is considered, where the matrix of a linear operator describing the system is an infinite matrix with $λ\in \mathbb R$ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if $λ\le-1$, which shows the fine difference between the finite and the infinite-dimensional systems. When $λ\le-1$ we also show that the system is null controllable in large. We also show a dependence of the stability on the norm i.e. the same system considered in $\ell^\infty$ is not asymptotically stable if $λ=-1$.

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arXiv2021

Linearized Korteweg -- De Vries equation on a tree with unbounded root and edges

We investigate the linearized KdV equation on a metric tree consisting of three different types of bonds: incoming unbounded root, two finite bonds, and four outgoing unbounded bonds. Under natural assumptions at the vertices, we obtain the uniqueness of a solution. To show the existence we use the theory of potentials and reduce the problem to a system of linear algebraic equations. We show that the latter is uniquely solvable under conditions of the uniqueness theorem. Also, we show that the system we consider can be used to model wave propagation in pipelines.

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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.

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arXiv2020

Variance continuity for Lorenz flows

The classical Lorenz flow, and any flow which is close to it in the $C^2$-topology, satisfies a Central Limit Theorem (CLT). We prove that the variance in the CLT varies continuously.

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arXiv2018

Almost sure rates of mixing for random intermittent maps

We consider a family $\mathcal F$ of maps with two branches and a common neutral fixed point $0$ such that the order of tangency at $0$ belongs to some interval $[α_0, α_1]\subset (0, 1)$. Maps in $\mathcal F$ do not necessarily share a common Markov partition. At each step a member of $\mathcal F$ is chosen independently with respect to the uniform distribution on $[α_0, α_1]$. We show that the construction of the random tower in Bahsoun-Bose-Ruziboev \cite{BBR} with \emph{general return time} can be carried out for random compositions of such maps. Thus their general results are applicable and gives upper bounds for the quenched decay of correlations of form $n^{1-1/α_0+δ}$ for any $δ>0$.

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arXiv2018

Linear response for random dynamical systems

We study for the first time linear response for random compositions of maps, chosen independently according to a distribution $\PP$. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when $\PP$ changes smoothly to $\PP_{\eps}$? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to $\eps$; moreover, we obtain a linear response formula. We apply our results to iid compositions, with respect to various distributions $\PP_{\eps}$, of uniformly expanding circle maps, Gauss-Rényi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

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arXiv2018

Quenched decay of correlations for slowly mixing systems

We study random towers that are suitable to analyse the statistics of slowly mixing random systems. We obtain upper bounds on the rate of quenched correlation decay in a general setting. We apply our results to the random family of Liverani-Saussol-Vaienti maps with parameters in $[α_0,α_1]\subset (0,1)$ chosen independently with respect to a distribution $ν$ on $[α_0,α_1]$ and show that the quenched decay of correlation is governed by the fastest mixing map in the family. In particular, we prove that for every $δ>0$, for almost every $ω\in [α_0,α_1]^\mathbb Z$, the upper bound $n^{1-\frac{1}{α_0}+δ}$ holds on the rate of decay of correlation for Hölder observables on the fibre over $ω$. For three different distributions $ν$ on $[α_0,α_1]$ (discrete, uniform, quadratic), we also derive sharp asymptotics on the measure of return-time intervals for the quenched dynamics, ranging from $n^{-\frac{1}{α_0}}$ to $(\log n)^{\frac{1}{α_0}}\cdot n^{-\frac{1}{α_0}}$ to $(\log n)^{\frac{2}{α_0}}\cdot n^{-\frac{1}{α_0}}$ respectively.

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arXiv2015

Young Towers for Product Systems

We show that the direct product of maps with Young towers admits a Young tower whose return times decay at a rate which is bounded above by the slowest of the rates of decay of the return times of the component maps. An application of this result, together with other results in the literature, yields various statistical properties for the direct product of various classes of systems, including Lorenz-like maps, multimodal maps, piecewise $ C^2 $ interval maps with critical points and singularities, Hénon maps and partially hyperbolic systems.

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