Maximally Chaotic Dynamical System of Infinite Dimensionality

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.