On the algebraic properties of exponentially stable integrable hamiltonian systems

Open Sets of Exponentially Mixing Anosov Flows

We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$, $\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable.This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$ uniforml…

非形式可积中心允许代数逆积分因子

Non-formally integrable centers admitting an algebraic inverse integrating factor

研究非形式可积系统逆积分因子的存在性并刻画其中心。

实连通半单矩阵群的代数向量场的可积生成元

Integrable Generators of Algebraic Vector Fields for Real Connected Semi-simple Matrix Groups

该研究证明实连通半单矩阵群的代数向量场的李代数可由三个完备向量场生成。

代数可积叶状结构的体积与局部稳定族

Volume of algebraically integrable foliations and locally stable families

本文研究代数可积叶状结构的体积与局部稳定族，证明典范代数可积叶状结构的体积属于仅依赖秩和一般叶体积的离散集，一般型时有正下界，并推导了稳定族相对体积的离散性及对数版本结果。

Persistence of Diophantine flows for quadratic nearly-integrable Hamiltonians under slowly decaying aperiodic time dependence

The aim of this paper is to prove a Kolmogorov-type result for a nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists of the possibility to choose an arbitrarily small decaying coefficient, consistently with the perturbation size.

On algebraic construction of certain integrable and super-integrable systems

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.