Nguyen Tien Zung
We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization. The proof is based on a new, geometric approach to the problem.
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The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillators with a homogeneous-{\it cubic} polynomial potential and with a homogeneous-{\it quartic} polynomial potentials admit the same BG-normalization up to degree-4 then both oscillators satisfy the Bertrand-Darboux integrability condition.
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral rigidity of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish rigidity for almost all of them.
We prove that a Hamiltonian $p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a non-degenerate critical point $ρ_0$ of the energy, provided that the fundamental matrix at $ρ_0$ has no purely imaginary eigenvalues. This is done by using Birkhoff normal forms, which turn out to be convergent in the $C^\infty$ sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the almost holomorphic case.
Birkhoff Normal Form and Twist Coefficients of Periodic Orbits of Billiards
本文针对多种动力台球，显式构造椭圆周期点附近的Birkhoff变换，并用台球桌几何参数给出前两个扭转系数的显式公式，进而刻画非线性稳定性与局部解析可积性。
On the Birkhoff conjecture for Kepler billiards
本文研究了开普勒台球系统的可积性，通过构建基于混沌子系统的符号动力学，证明了高能量下解析可积性仅在椭圆区域且引力中心位于焦点时成立，为 Birkhoff-Poritsky 猜想提供了部分肯定回答。
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in [1], where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the cruci…
We establish a generic sufficient condition for a compact $n$-dimensional manifold bearing an integrable geodesic flow to be the $n$-torus. As a complementary result, we show that in the case of domains of possible motions with boundary, the first Betti number of the domain of possible motions may be arbitrarily large.
The purpose of this article is to show that on an open and dense set, complete integrability implies the existence of symmetry.
We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies the existence of a positive Lebesgue measure subset of points with positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic ti…
We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic times at Lebesgue almost every point but whose first hyperbolic time …
Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards
本文研究Kustaanheimo-Stiefel变换与Birkhoff-Waldvogel变换在三维开普勒台球中的应用，揭示可积反射壁的旋转不变性，并建立Levi-Civita平面理论证明双中心台球的可积性。
We show that, to find a Poincare-Dulac normalization for a vector field is the same as to find and linearize a torus action which preserves the vector field. Using this toric characterization and other geometrical arguments, we prove that any local analytic vector field which is integrable in the non-Hamiltonian sense admits a local convergent Poincare-Dulac normalization. These results generalize the main results of our previous paper from the Hamiltonian case to the non-Hamiltonian case. Similar results are presented for the case of isochore vector fields.
Among other results we show that near the equilibrium point, the Hamiltonian of the sine-Gordon (SG) equation on the circle can be viewed as an element in the Poisson algebra of the modified Korteweg-de Vries (mKdV) equation and hence by well established properties of the latter equation admits Birkhoff coordinates. On the other hand we prove that there exists a large set of smooth initial data, away from the equilibrium point and lying on the ramification locus of a double cover, for which the initial value problem of the SG equation has no classical solution.
We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schr{ö}dinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant near $\hbar= 0$. The unperturbed frequencies s…
Integrable Birkhoff Billiards inside Cones
本文证明$\mathbb{R}^n$中凸$C^3$锥体内的Birkhoff台球具有二次第一积分且轨迹反射次数有限，并首次给出既非二次曲面也非二次曲面组合的可积台球例子。
In 1968, V.I. Oseledets formulated the question of convergence in the Birkhoff theorem and the multiplicative ergodic theorem for measurable cocycles over flows under the condition of integrability for each individual t. A.M. Stepin and the author established (2016) the convergence along subsets of time of density 1. In this note, we show that moreover the convergence is fulfilled modulo time subsets of finite measure.
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