Leboeuf P
Laboratoire de Physique Théorique et Modèles Statistiques, Université de Paris XI, Bâtiment 100, 91405 Orsay Cedex, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Feb;69(2 Pt 2):026204. doi: 10.1103/PhysRevE.69.026204. Epub 2004 Feb 25.
Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g., a trajectory "p" returns to its initial conditions after some fixed time tau(p). Our aim is to investigate the spectrum [tau(1),tau(2), ...] of periods of the periodic orbits. An explicit formula for the density rho(tau)= Sigma(p)delta(tau-tau(p)) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle-Pollicott resonances). For large periods, corrections to the well-known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random-matrix theory, and discrete maps are also considered. In particular, a random-matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.
完全混沌的哈密顿系统拥有无限多个周期的经典解，例如，一条轨迹“p”在经过某个固定时间τ(p)后会回到其初始条件。我们的目标是研究周期轨道的周期谱[τ(1),τ(2),...]。根据经典演化算符的本征值，推导出了周期密度ρ(τ)=Σ(p)δ(τ - τ(p))的显式公式。该密度自然地分解为一个光滑部分加上一个关于振荡项的干涉和。振荡项的频率由复本征值的虚部给出（吕埃勒 - 波利科特共振）。对于大周期，得到了对密度光滑部分著名的指数增长的修正。还讨论了根据吕埃勒ζ函数的零点和极点得到的ρ(τ)的另一种公式。结果通过常负曲率台球中的测地线运动进行了说明。还考虑了与相应量子本征值的统计性质、随机矩阵理论和离散映射的联系。特别是，针对混沌台球的经典演化算符的本征值提出了一个随机矩阵猜想。
