How well can one resolve the state space of a chaotic map?

Abstract

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. We propose to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. Numerical tests of such `optimal partition' of a one-dimensional repeller support our hypothesis.