双曲Anosov C系统关联函数的指数衰减
Hyperbolic Anosov C-systems. Exponential Decay of Correlation Functions
摘要：
定义在环面上的一致双曲Anosov C-系统具有其轨迹的指数不稳定性，因此C-系统具有所有阶的混合和非零Kolmogorov熵。所有阶的混合特性意味着它的所有关联函数趋于零，并且基本感兴趣的问题是它们趋于零的速度。证明了C-系统的衰减速度是指数的，即相空间上的观测量以指数的速度变得独立和不相关。重要的是要指定的C-系统的属性，量化的指数衰减的相关性。我们发现，关联函数指数衰减的上界普遍依赖于系统熵的值。分析的精髓是，局部和均匀的不稳定性的C-系统相空间轨迹转化为指数衰减的关联函数的速率是成比例的Kolmogorov熵，Anosov自同构的基本特征之一。这一结果允许定义的解相关和弛豫时间的C-系统的熵和统计性质的广泛的一类动力系统，包括伪随机数发生器和引力系统。
英文摘要：
The uniformly hyperbolic Anosov C-systems defined on a torus have exponential instability of their trajectories, and as such C-systems have mixing of all orders and nonzero Kolmogorov entropy. The mixing property of all orders means that all its correlation functions tend to zero and the question of a fundamental interest is a speed at which they tend to zero. It was proven that the speed of decay in the C-systems is exponential, that is, the observables on the phase space become independent and uncorrelated exponentially fast. It is important to specify the properties of the C-system which quantify the exponential decay of correlations. We have found that the upper bound on the exponential decay of the correlation functions universally depends on the value of a system entropy. A quintessence of the analyses is that local and homogeneous instability of the C-system phase space trajectories translated into the exponential decay of the correlation functions at the rate which is proportional to the Kolmogorov entropy, one of the fundamental characteristics of the Anosov automorphisms. This result allows to define the decorrrelation and relaxation times of a C-system in terms of its entropy and characterise the statistical properties of a broad class of dynamical systems, including pseudorandom number generators and gravitational systems.
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