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We analyse the infinite-dimensional limit of the maximally chaotic dynamicalsystems that are defined on N-dimensional tori. These hyperbolic systems foundsuccessful application in computer algorithms that generate high-qualitypseudorandom numbers for advanced Monte Carlo simulations. The chaoticproperties of these systems are increasing with $N$ because the correspondingKolmogorov-Sinai entropy grows linearly with $N$. We calculated the spectrumand the entropy of the system that appears in the infinite dimensional limit.We demonstrated that the limiting system has exponentially expanding andcontracting foliations and therefore belongs to the Anosov maximally chaoticdynamical systems of infinite dimensionality. The liming system defines thehyperbolic evolution of the continuous functions very similar to the evolutionof a velocity function describing the hydrodynamic flow of fluids. We comparethe chaotic properties of the limiting system with those of the hydrodynamicflow of incompressible ideal fluid on a torus investigated by Arnold. Thismaximally chaotic system can find application in Monte Carlo method,statistical physics and digital signal processing.

Maximally Chaotic Dynamical System of Infinite Dimensionality