Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
复制标题
较高属表面上保面积流的遍历平均值的偏差
DOI：
10.2307/3062150
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发表时间：
2004
影响因子：
4.9
通讯作者：
G. Forni
中科院分区：
文献类型：
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作者：
G. Forni
我们证明了相当大的一部分猜想Kontsevich和Zorich的Teichmuller测地线流的Lyapunov指数的偏差一般保守流的遍历平均更高的属表面。Teichmuller流的结果是在由Kontsevich和Zorich引入的全纯微分模空间上的真实的上同调丛上的（辛）上循环的形式。我们证明了这样的上圈是非一致双曲的，即它的所有李雅普诺夫指数都不为零。特别地，严格正指数的数目等于曲面的亏格。从这个定理中，我们得出，遍历积分的光滑函数的一般面积保持流在较高的亏格表面上的增长与时间根据幂律渐近的一些条款等于亏格的表面和严格的正指数等于非负的李雅普诺夫指数的Kontsevich-Zorich上循环。特别是，保守流表面上的高亏格，遍历平均的一般光滑函数的偏差服从幂律与严格的正指数，因此，Denjoy-Koksma不等式不成立。偏差定理的推导从根本上依赖于表面上流动的不变分布概念和轨道叶状的基本电流的相关概念。
We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces. The result on the Teichmuller flow is formulated in terms of a (symplectic) cocycle on the real cohomology bundle over the moduli space of holomorphic differentials introduced by Kontsevich and Zorich. We prove that such a cocycle is non-uniformly hyperbolic, that is, all of its Lyapunov exponents are different from zero. In particular, the number of strictly positive exponents is equal to the genus of the surface. From this theorem we derive that ergodic integrals of smooth functions for generic area-preserving flows on higher genus surfaces grow with time according to a power-law asymptotics with a number of terms equal to the genus of the surface and stricltly positive exponents equal to the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle. In particular, for conservative flows on surfaces of higher genus, the deviation of ergodic averages for a generic smooth function obeys a power law with a strictly positive exponent and, consequently, the Denjoy-Koksma inequality does not hold. The derivation of the deviation theorem relies in a fundamental way on the notion of invariant distribution for flows on surfaces and the related notion of basic current for the orbit foliation.