COARSE-GRAINED OBSERVATION OF DISCRETIZED MAPS

Abstract

We investigate why discretized versions fN of one-dimensional ergodic maps f : I → I behave in many ways similarly to their continuous counterparts. We propose to register observations of the N × N discretization fN on a coarse M × M grid, with N = cM , c being an integer. We prove that rounding errors behave like uniformly distributed random variables, and by assuming their independence, the M ×M incidence matrix A associated with the continuous map (indicating which of the M equal subintervals is mapped onto which) can be expected to be identical to the incidence matrix B associated with the aforementioned coarse grid, if c ≥ √deg(f)N , where deg(f) denotes the degree of f . We show how coarse-grained registration can be used as a“digital” definition of an unstable orbit and how this can be applied in real computations. Combination of these results with ideas from the random map model suggests an intuitive explanation for the statistical similarity between f and fN . Our approach is not a rigorous one, however, we hope that the results will be useful for the computational community and may facilitate a rigourous mathematical description.