Maximally Chaotic Dynamical System of Infinite Dimensionality

TL;DR

This paper investigates the infinite-dimensional limit of maximally chaotic systems on tori, revealing their hyperbolic structure and potential applications in Monte Carlo methods, physics, and signal processing.

Contribution

It introduces the infinite-dimensional limit of maximally chaotic hyperbolic systems and analyzes their spectral and entropy properties, establishing their relation to hydrodynamic flows.

Findings

The system's Kolmogorov-Sinai entropy grows linearly with dimension.

The limiting system exhibits exponential expansion and contraction, characteristic of Anosov systems.

It models hyperbolic evolution of functions similar to fluid dynamics.

Abstract

We analyse the infinite-dimensional limit of the maximally chaotic dynamical systems that are defined on N-dimensional tori. These hyperbolic systems found successful application in computer algorithms that generate high-quality pseudorandom numbers for advanced Monte Carlo simulations. The chaotic properties of these systems are increasing with $N$ because the corresponding Kolmogorov-Sinai entropy grows linearly with $N$. We calculated the spectrum and the entropy of the system that appears in the infinite dimensional limit. We demonstrated that the limiting system has exponentially expanding and contracting foliations and therefore belongs to the Anosov maximally chaotic dynamical systems of infinite dimensionality. The liming system defines the hyperbolic evolution of the continuous functions very similar to the evolution of a velocity function describing the hydrodynamic flow of fluids. We compare the chaotic properties of the limiting system with those of the hydrodynamic flow of incompressible ideal fluid on a torus investigated by Arnold. This maximally chaotic system can find application in Monte Carlo method, statistical physics and digital signal processing.