Related papers: Quantitative KAM normal forms and sharp measure es…
Particle tracking codes are one of the fundamental tools used in the design and the study of complex magnetic lattices in accelerator physics. For most practical applications, non-linear lenses are included and the Courant-Snyder formalism…
The first author and Latr\'emoli\`ere had introduced a quantum metric (in the sense of Rieffel) on the algebra of complex-valued continuous functions on the Cantor space. We show that this quantum metric is distinct from the quantum metric…
We review Kolmogorov's 1954 fundamental paper {\sl On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian} (Dokl. akad. nauk SSSR,1954, vol. {\bf 98}, pp.527--530), both from the historical and the…
We prove a new invariant torus theorem, for $\alpha$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $\alpha$-Bruno-R\"ussmann condition, and which reduces to the classical Bruno-R\"ussmann condition in…
We prove the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. This result is…
The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying mechanism for the stability of the solar system under some small chaotic perturbations. Despite many previous efforts, any quantum version of the KAM theorem…
We work with small non-selfadjoint perturbations of a selfadjoint quantum Hamiltonian with two degrees of freedom, assuming that the principal symbol of the selfadjoint part is (classically) a nearly integrable system, together with a…
The disadvantage of `traditional' multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis…
By constructing an infinite dimensional KAM theorem of the normal frequencies being dense at finite-point, we show that some shallow water equations such as Benjamin-Bona-Mahony equation and the generalized $d$-Dim. Pochhammer-Chree…
We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on…
In 1967 Moser proved the existence of a normal form for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. In this paper we present a proof of existence of this normal form based…
The persistence of invariant tori in multi-scale Hamiltonian systems is intrinsically linked to the stability of the N-body problem. However, the existing non-degeneracy conditions in disordered scenarios have been formulated too generally,…
We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman's method. The reversible KAM context 2 refers to the situation where the…
In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schr\"odinger equation with periodic boundary conditions \begin{equation*}\label{L1}…
In the Nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophatine critical point of a hamiltonian function. I construct a formalism for the UV-cutoff and prove a generalised KAM…
We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance…
We prove a reducibility result for a linear Klein-Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving, however we require it to be fast…
We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact…
In this paper, we introduce a novel and generic approach to prove the persistence of frequency-preserving invariant tori in parameterized Hamiltonian systems, addressing irregular continuity with respect to parameters. Unlike traditional…
A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we…