arXiv subjects

Maxence Phalempin

Publications and source records attributed to Maxence Phalempin.

7 recordsLinked to original sources

arXiv2026

Annealed Ruelle-Pollicott Resonances

We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.

math.DS↗

arXiv2026

Optimal linear response for Anosov diffeomorphisms

It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $\dot{T}$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gouëzel--Liverani. We prove that the optimising perturbation $\dot{T}$ is unique for non-degenerate response functions and provide explicit expressions for the Fourier coefficients of $\dot{T}$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $\dot{T}$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations.

math.DS↗

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.

math.DS↗

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 $\textbf{p}^*$.

math.DS↗

arXiv2024

Slow-fast systems in infinite measure, with or without averaging

This paper studies the asymptotic behaviour of the solution of a differential equation perturbed by a fast flow preserving an infinite measure. This question is related with limit theorems for non-stationary Birkhoff integrals. We distinguish two settings with different behaviour: the integrable setting (no averaging phenomenon) and the case of an additive "centered" perturbation term (averaging phenomenon). The paper is motivated by the case where the perturbation comes from the Z-periodic Lorentz gas flow or from the geodesic flow over a Z-cover of a negatively curved compact surface. We establish limit theorems in more general contexts.

math.DS↗

arXiv2024

Averaging theorems for slow fast systems in $\mathbb{Z}$-extensions (discrete time)

We study the averaging method for flows perturbed by a dynamical system preserving an infinite measure. Motivated by the case of perturbation by the collision dynamic on the finite horizon $\mathbb Z$-periodic Lorentz gas and in view of future development, we establish our results in a general context of perturbation by $\mathbb Z$-extension over chaotic probability preserving dynamical systems. As a by product, we prove limit theorems for non-stationary Birkhoff sums for such infinite measure preserving dynamical systems.

math.DS↗

arXiv2023

An alternative approach to large deviations for the almost-critical Erdős-Rényi random graph

We study the near-critical behavior of the sparse Erdős-Rényi random graph $\mathcal{G}(n,p)$ on $n\gg1$ vertices, where the connection probability $p$ satisfies $np = 1+θ(b_n^2/n)^{1/3}$, with $n^{3/10}\ll {b_n}\ll n^{1/2}$, and $θ\in\mathbb{R}$. To this end, we introduce an empirical measure that describes connected components of $\mathcal{G}(n,p)$ of mesoscopic size $\propto (nb_n)^{2/3}$, and we characterize its large deviation behavior. The proof hinges on detailed combinatorial estimates and optimization procedures. In particular, we give precise estimates for the probability that the graph has no connected component of mesoscopic size or larger. We argue that these are a stepping stone for the analysis of more general inhomogeneous random graphs. Our proof strategy gives new and accurate estimates of the probability that the sparse Erdős-Rényi graph is connected.

math.PR↗