论文标题
无限可观察的相关性衰减
Decay of Correlations for Unbounded Observables
论文作者
论文摘要
在本文中，我们研究双曲线系统的相关函数的衰减率$ t：m \至m $具有奇异性，可保留独特的混合SRB测量$μ$。我们证明，在某些一般假设下，相关性$ c_ {n}（f，g）$衰减为$ n \ to \ to \ infty $，每对piewisehölderobservables $ f，g \ in l^p（μ）$，每个$ p> 1 $。作为一个应用程序，我们证明了第一个返回时间功能的自相关对各种台球系统的诱导地图成倍衰减，其中包括矩形上的半分散台球，带有cusps的台球和Bunimovich STADIA（用于第一个回返回时间功能）。这些诱导地图的第一个回返回时间函数的自相关衰减率的估计值在研究非均匀双曲线系统（具有奇异性）的统计特性方面至关重要。
In this article, we study the decay rates of the correlation functions for a hyperbolic system $T: M \to M$ with singularities that preserves a unique mixing SRB measure $μ$. We prove that, under some general assumptions, the correlations $ C_{n}(f,g)$ decay exponentially as $n\to \infty$ for each pair of piecewise Hölder observables $f, g\in L^p(μ)$ and for each $p>1$. As an application, we prove that the autocorrelations of the first return time functions decay exponentially for the induced maps of various billiard systems, which include the semi-dispersing billiards on a rectangle, billiards with cusps, and Bunimovich stadia (for the truncated first return time functions). These estimates of the decay rates of autocorrelations of the first return time functions for the induced maps have an essential importance in the study of the statistical properties of nonuniformly hyperbolic systems (with singularities).
