Frequency measurability, algebras of quasiperiodic sets and spatial discretizations of smooth dynamical systems

The roundoff errors in computer simulations of continuous dynamical systems caused by finiteness of machine arithmetic lead to spatially discrete systems with distinct functional properties. As models for the discretized systems in fixed-point arithmetic, autonomous dynamical systems on multi-dimensional uniform grids can be considered which are generated by composing the transition operator of the original system with a roundoff mapping. To study asymptotic properties of such model systems with increasing refinement of the grid, a probability theoretical approach is developed which is based on equipping the grid with an algebra of frequency-measurable quasiperiodic subsets characterized by frequency functions. The approach is applied to spatial discretizations of smooth and invertible dynamical systems. It is shown that, under some nonresonance condition, events relating to mutual deviations of finite segments of trajectories of the discretized and original systems can be represented by frequency-measurable quasiperiodic sets with frequency functions amenable to explicit computation.