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

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