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Hyperbolic Anosov C-systems. Exponential Decay of Correlation Functions
Mathematical Physics · Submitted 12 Feb 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.07 | 0.06 | asserted 1.00 |
| s1 | 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. | background 0.86 | 0.06 | 0.04 | asserted 1.00 |
| s2 | 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. | background 0.88 | 0.07 | 0.20 | asserted 1.00 |
| s3 | It is important to specify the properties of the C-system which quantify the exponential decay of correlations. | background 0.51 | 0.10 | 0.04 | asserted 0.57 |
| s4 | We have found that the upper bound on the exponential decay of the correlation functions universally depends on the value of a system entropy. | result 0.96 | 0.94 | 0.69 | asserted 1.00 |
| s5 | 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. | interpretation 0.67 | 0.80 | 0.85 | asserted 1.00 |
| s6 | 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. | interpretation 0.99 | 0.73 | 0.32 | asserted 0.96 |
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The paper
Original 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.
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How the literature uses it
2 arXiv papers cite this in 9 sentences
Papers citing it · 6 sentences without a confident label
Cites it as support 1 paper
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The MIXMAX generator of pseudorandom numbers represents the homogeneous C-system for which the classical limit theorems are fulfilled [ref] .
Classical limit theorems and high entropy MIXMAX random number generator · Introduction · 1.00
Cites it as background 1 paper
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The hyperbolic Anosov C-systems defined on a torus have mixing of all orders and nonzero Kolmogorov entropy [ref] .
Classical limit theorems and high entropy MIXMAX random number generator · Classical Limit Theorems and Deterministic C-systems · 0.87
What it cites · 29 sentences without a confident label
Uses data or values from 1 paper
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The generator $N=256$ in Table 1 of the article [ref] has the entropy $h(T)=194$ and the smallest phase volume is of order $δv_0 = 2^- 61 ·256$, therefore the characteristic time scales for this generator are [ Decorrelation time; τ_0 =0.000012 ] < [ Interaction time; t_int =1 ] < [ Stationary distribution time; τ=95 ].
Spectrum and Entropy of C-systems. MIXMAX random number generator · MIXMAX Random Number Generator · 0.95
Cites as support 4 papers
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It was proven that for the hyperbolic Anosov C-systems the speed of decay is exponential, that is, the observables on the phase space become independent and uncorrelated exponentially fast [ref] .
Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances · Introduction · 0.96
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It was proven that for the hyperbolic Anosov C-systems the speed of decay is exponential, that is, the observables on the phase space become independent and uncorrelated exponentially fast [ref] .
Quasi-compactness of transfer operators for contact Anosov flows · Introduction · 0.96
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It was proven that for the hyperbolic Anosov C-systems the speed of decay is exponential, that is, the observables on the phase space become independent and uncorrelated exponentially fast [ref] .
Upper bound on the density of Ruelle resonances for Anosov flows · Introduction · 0.96
Cites as background 2 papers
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The earlier publications concerning the application of the modern results of the ergodic theory to concrete physical systems can be found in [ref] .
The Jacobi-metric for timelike geodesics in static spacetimes · Introduction · 1.00
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It was suggested in the literature that the outgoing Hawking radiation [ref] may be not exactly thermal, but had subtle correlations [ref] .
Chaos in Classical D0-Brane Mechanics · Conclusion · 0.91
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