  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.
