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Creators/Authors contains: "Fayad, Bassam"

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  1. ; ; ; (, Nonlinearity)
    Abstract We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles is asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies. 
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    Free, publicly-accessible full text available April 11, 2026
  2. ; ; (, Discrete and Continuous Dynamical Systems)
    We give an example of a real analytic reparametrization of a minimal translation flow on $$\mathbb{T}^{5}$$ that has a Lebesgue spectrum with infinite multiplicity. As a consequence, we see that the dynamics on a non-Diophantine invariant torus of an almost integrable Hamiltonian system can be spectrally equivalent to a Bernoulli flow. 
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  3. (, Journal of the American Mathematical Society)
    A new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom is introduced. We thus obtain explicit real entire Hamiltonians on R 2 d \mathbb {R}^{2d} , d ≥ 4 d\geq 4 , that have a Lyapunov unstable elliptic equilibrium with an arbitrary chosen frequency vector whose coordinates are not all of the same sign. For non-resonant frequency vectors, our examples all have divergent Birkhoff normal form at the equilibrium. On R 4 \mathbb {R}^4 , we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form. 
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  4. ; (, Journal of the European Mathematical Society)
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  5. ; (, Ergodic Theory and Dynamical Systems)
    Abstract We construct a $C^1$ symplectic twist map g of the annulus that has an essential invariant curve $$\Gamma $$ such that $$\Gamma $$ is not differentiable and g restricted to $$\Gamma $$ is minimal. 
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  6. ; (, Ergodic Theory and Dynamical Systems)
    Abstract We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies. 
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  7. ; (, Discrete & Continuous Dynamical Systems)
    Any \begin{document}$ C^d $$\end{document} conservative map \begin{document}$$ f $$\end{document} of the \begin{document}$$ d $$\end{document}-dimensional unit ball \begin{document}$$ {\mathbb B}^d $$\end{document}, \begin{document}$$ d\geq 2 $$\end{document}, can be realized by renormalized iteration of a \begin{document}$$ C^d $$\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}$$ {\mathbb B}^d $$\end{document}, arbitrarily close to identity in the \begin{document}$$ C^d $$\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$$ C^d $$\end{document} change of coordinates is exactly \begin{document}$$ f $$\end{document}$. 
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  8. ; ; (, Journal of Modern Dynamics)
    A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems. 
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  9. ; ; (, Journal of the American Mathematical Society)
    We study the spectral measures of conservative mixing flows on the 2 2 -torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: (1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, (2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows. 
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  10. ; ; (, Electronic Journal of Probability)
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