Spectrum and Entropy of C-systems. MIXMAX random number generator

Abstract

The uniformly hyperbolic Anosov C-systems defined on a torus have very strong instability of their trajectories, as strong as it can be in principle. These systems have exponential instability of all their trajectories and as such have mixing of all orders, nonzero Kolmogorov entropy and a countable set of everywhere dense periodic trajectories. In this paper we are studying the properties of their spectrum and of the entropy. For a two-parameter family of C-system operators A(N,s), parametrised by the integers N and s, we found the universal limiting form of the spectrum, the dependence of entropy on N and the period of its trajectories on a rational sublattice. One can deduce from this result that the entropy and the periods are sharply increasing with N. We present a new three-parameter family of C-operators A(N,s,m) and analyse the dependence of its spectrum and of the entropy on the parameter m. We developed our earlier suggestion to use these tuneable Anosov C-systems for multipurpos Monte-Carlo simulations. The MIXMAX family of random number generators based on Anosov C-systems provide high quality statistical properties, thanks to their large entropy, have the best combination of speed, reasonable size of the state, tuneable parameters and availability for implementing the parallelisation.

12 pages, 5 figures, references, comments and a note added

References in corpus (3)

• Recurrence Plots for the Analysis of Complex Systems
• Anosov C-systems and random number generators
• Identification of Two Radio Loud Narrow Line Seyfert 1 Galaxies at Gamma-Ray Energies

Cited by in corpus (10)

• Spectral Test of the MIXMAX Random Number Generators
• Maximally Chaotic Dynamical Systems and Fundamental Interactions
• Artin Billiard Exponential Decay of Correlation Functions
• Distribution of periodic trajectories of Anosov C-system
• Maximally Chaotic Dynamical System of Infinite Dimensionality
• Extended Kolmogorov Entropy
• Maximally chaotic dynamical systems of Anosov-Kolmogorov
• Yang-Mills Classical and Quantum Mechanics and Maximally Chaotic Dynamical Systems
• Statistical tests for MIXMAX pseudorandom number generator
• Classical limit theorems and high entropy MIXMAX random number generator