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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 nonresonant 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 nonresonant frequency vector and a divergent Birkhoff normal form.more » « less

Any
conservative map\begin{document}$ C^d $\end{document} of the\begin{document}$ f $\end{document} dimensional unit ball\begin{document}$ d $\end{document} ,\begin{document}$ {\mathbb B}^d $\end{document} , can be realized by renormalized iteration of a\begin{document}$ d\geq 2 $\end{document} perturbation of identity: there exists a conservative diffeomorphism of\begin{document}$ C^d $\end{document} , arbitrarily close to identity in the\begin{document}$ {\mathbb B}^d $\end{document} topology, that has a periodic disc on which the return dynamics after a\begin{document}$ C^d $\end{document} change of coordinates is exactly\begin{document}$ C^d $\end{document} .\begin{document}$ f $\end{document} 
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.

We study the spectral measures of conservative mixing flows on the
$2$ 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 nonuniformly 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.