Distribution of periodic trajectories of Anosov C-system

TL;DR

This paper investigates the exact distribution and deviations of periodic trajectories in hyperbolic Anosov C-systems, especially those used in pseudorandom number generation, with a focus on high-dimensional tori.

Contribution

It provides new insights into the distribution deviations of periodic trajectories in hyperbolic C-systems, including the implementation in the MIXMAX pseudorandom number generator.

Findings

Exact distribution of periodic trajectories studied

Deviation from asymptotic behavior analyzed

Application to the MIXMAX generator at CERN

Abstract

The hyperbolic Anosov C-systems have a countable set of everywhere dense periodic trajectories which have been recently used to generate pseudorandom numbers. The asymptotic distribution of periodic trajectories of C-systems with periods less than a given number is well known, but a deviation of this distribution from its asymptotic behaviour is less known. Using fast algorithms, we are studying the exact distribution of periodic trajectories and their deviation from asymptotic behaviour for hyperbolic C-systems which are defined on high dimensional tori and are used for Monte-Carlo simulations. A particular C-system which we consider in this article is the one which was implemented in the MIXMAX generator of pseudorandom numbers. The generator has the best combination of speed, reasonable size of the state, and availability for implementing the parallelization and is currently available generator in the ROOT and CLHEP software packages at CERN.