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We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian}, and prove that, optimizing Arnold's scheme, one can get "sharp" asymptotic quantitative conditions (as $\varepsilon\to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed., Comment: To appear in

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori conjecture namely the existence of a positive measure set of invariant tori near the critical point. This paper is an update of the first 2012 proof of the author. The functional analytic arguments have been simplified using Banach functors, minor points have been clarified. A series of videos is available on the webpage https://www.agtz.mathematik.uni-mainz.de/category/alg-geom/

Chow, Li and Yi in [2] proved that the majority of the unperturbed tori {\it on sub-manifolds} will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type., 25 pages. To appear in Nonlinear Analysis: TMA

We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has to satisfy in order for the perturbed Hamiltonian to be a first order deformation., Comment: minor typo errors fixed, bibliography updated

Let $R_1, R_2,\ldots,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$ , $d\geq 2$ , and let $f_1, f_2,\ldots,f_m$ be small smooth perturbations of them. We show that $\{f_\alpha\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions

We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the theorem presented here is to provide exactly the estimates needed in [1]., Comment: 48 pages

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining $C^\infty$ functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in the case of two-dimensional rotation vectors with Diophantine exponent $\tau=1$ (e. g. with ratio of the two independent frequencies equal to the golden mean)., Comment: 17 pages, 1 figure

Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple) normal $1:-1$ resonance. This theory guarantees the persistence of the invariant torus in the Diophantine case and makes possible a further quasi-periodic normal form, necessary for investigation of the non-linear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation., Comment: Published at http://dx.doi.org/10.1214/074921706000000301 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

We propose a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m, m\geq 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of $n$ non-linear rotators via a small analytic potential. The global bifurcation problem is set-up for the $n$-dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on an $n$-torus, whose $k$th Fourier coefficient satisfies the estimate $$O(e^{- \rho|k\cdot\omega| - |k|\sigma}), k\in\Z^n\setminus\{0\},$$ where $\omega\in\R^n$ is a Diophantine rotation vector of the system of rotators; $\rho\in(0,{\pi\over2})$ and $\sigma>0$ are the analyticity parameters built into the model. The estimate, under suitable assumptions would generalize to a general multiple resonance normal form of a convex analytic Liouville integrable Hamiltonian system, perturbed by $O(\eps)$, in which case $\omega_j\sim\omeps, j=1,...,n.$, Comment: 24 pages