Hyperbolic Anosov C-systems. Exponential Decay of Correlation Functions

TL;DR

This paper proves that hyperbolic Anosov C-systems on a torus exhibit exponential decay of correlation functions, with the decay rate universally bounded by the system's entropy, linking dynamical instability to statistical independence.

Contribution

It establishes a universal bound on the exponential decay rate of correlations in Anosov C-systems based on their entropy, connecting dynamical instability with statistical properties.

Findings

Correlation functions decay exponentially fast.

Decay rate is proportional to the system's Kolmogorov entropy.

Results apply to pseudorandom generators and gravitational systems.

Abstract

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.