Mapping densities in a noisy state space

Weak noise smooths out fractals in a chaotic state space and introduces a maximum attainable resolution to its structure. The balance of noise and deterministic stretching/contraction in each neighborhood introduces local invariants of the dynamics that can be used to partition the state space. We study the local discrete-time evolution of a density in a two-dimensional hyperbolic state space, and use the asymptotic eigenfunctions for the noisy dynamics to formulate a new state space partition algorithm.