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Wael Bahsoun

Publications and source records attributed to Wael Bahsoun.

At least 19 recordsLinked to original sources
arXiv2026

Decay of Correlations for Partially Hyperbolic Skew-Products

We develop a technique based on sectional transfer operators and the regularity of their fixed points to study rates of correlation decay for partially hyperbolic skew-products. To illustrate the range of applicability of this technique, we apply it to a family of Viana like maps with non-uniformly contracting fibres, a family of skew-product maps with non-uniformly expanding base dynamics and to a generalised family of heterochaos baker maps. The latter partially hyperbolic systems have recently been an active topic of research due to their connection with the Dyck shift.

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arXiv2026

Fixed-point approximation for self-consistent transfer operators with Newton's method

Self consistent transfer operators arise naturally in the study of mean-field coupled dynamical systems and are closely related to kinetic PDEs such as the Vlasov equation. Despite substantial progress on existence and uniqueness of fixed points for self-consistent transfer operators, the development of fast, reliable, and provably accurate numerical methods remains largely unresolved. In this work, we construct a nonlinear Fourier-Fejér discretisation and establish convergence of the resulting finite-dimensional fixed point to that of the original self-consistent transfer operator. Further, using the nonlinear Fourier-Fejér discretisation, we prove exponential convergence of a sequential iteration scheme and develop a Newton framework with quadratic convergence. We present numerical examples demonstrating the efficiency and flexibility of the above methods.

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arXiv2026

Linear response for skew-product maps with contracting fibres

We study linear response for families of skew-product dynamical systems with contracting fibres. Our approach is based on a sectional transfer operator acting on families of probability measures along the fibres. The operator allows to describe invariant measures of the skew-product in terms of sample measures over the base dynamics, regardless of invertibility or non-invertibility of the base map. Under general assumptions, we establish existence and uniqueness of invariant sample measures and prove their differentiability, with respect to system parameters, in suitable topologies. As an application we obtain linear response for Bernoulli convolutions, which are of prime importance in the study of number theoretic problems and fractals. Another application of our results yields linear response for physical measures of solenoidal attractors with intermittency, an example of a hyperbolic system which cannot be handled by traditional transfer operator techniques.

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arXiv2026

Rare events statistics for $\mathbb Z^d$ map lattices coupled by collision

Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate $\mathbb Z^d$-map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site $\textbf{p}^*\in \mathbb Z^d$ and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site $\textbf{p}^*$ converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site $\textbf{p}^*$.

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arXiv2025

Mean field coupled dynamical systems: Bifurcations and phase transitions

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses that are tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled expanding maps to globally coupled Axiom A diffeomorphisms. To illustrate the range of applicability, we analyze an explicit example consisting of globally coupled Anosov diffeomorphisms. For such an example, we classify all the invariant measures as the coupling strength varies; we show which invariant measures are physical, and we prove that the existence of multiple invariant physical measures is a purely infinite dimensional phenomenon, i.e., the model exhibits phase transitions in the sense of statistical mechanics.

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arXiv2024

Linear response due to singularities

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a \emph{cusp} at the turning point, we recover the linear response. More precisely, let $T_\eps$ be a family of such cusp maps generated by changing the value of the turning point of $T_0$ by a deterministic perturbation and let $h_\eps$ be the corresponding invariant density. We prove that $\eps\mapsto h_\eps$ is differentiable in $L^1$ and provide a formula for its derivative.

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arXiv2024

Impulsive Lorenz semiflows: Physical measures, statistical stability and entropy stability

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures of the semiflows are close, in the weak* topology, to the unique physical measure of the Lorenz flow. A similar conclusion holds for the entropies associated with the physical measures.

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arXiv2023

Statistical aspects of mean field coupled intermittent maps

We study infinite systems of mean field weakly coupled intermittent maps in the Pomeau-Manneville scenario. We prove that the coupled system admits a unique ``physical'' stationary state, to which all absolutely continuous states converge. Moreover, we show that suitably regular states converge polynomially.

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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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arXiv2022

Globally coupled Anosov diffeomorphisms: Statistical properties

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous.

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arXiv2022

Map lattices coupled by collisions: hitting time statistics and collisions per lattice unit

We study map lattices coupled by collision and show how perturbations of transfer operators associated with the spatially periodic approximation of the model can be used to extract information about collisions per lattice unit. More precisely, we study a map on a finite box of $L$ sites with periodic boundary conditions, coupled by collision. We derive, via a non-trivial first order approximation for the leading eigenvalue of the rare event transfer operator, a formula for the first collision rate and a corresponding first hitting time law. For the former we show that the formula scales at the order of $L\cdot\varepsilon^2$, where $\varepsilon$ is the coupling strength, and for the latter, by tracking the $L$ dependency in our arguments, we show that the error in the law is of order $O\left(C(L)\frac{L\varepsilon^2}{ζ(L)}\cdot\left|\ln \frac{L\varepsilon^2}{ζ(L)}\right|\right)$, where $ζ(L)$ is given in terms of the spectral gap of the rare event transfer operator, and $C(L)$ has an explicit expression. Finally, we derive an explicit formula for the first collision rate per lattice unit.

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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

Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.

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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

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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arXiv2017

A Rigorous Computational Approach to Linear Response

We present a general setting in which the formula describing the linear response of the physical measure of a perturbed system can be obtained. In this general setting we obtain an algorithm to rigorously compute the linear response. We apply our results to expanding circle maps. In particular, we present examples where we compute, up to a pre-specified error in the $L^{\infty}$-norm, the response of expanding circle maps under stochastic and deterministic perturbations. Moreover, we present an example where we compute, up to a pre-specified error in the $L^1$-norm, the response of the intermittent family at the boundary; i.e., when the unperturbed system is the doubling map.

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