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Application of ISOMAP in Dynamical Systems , Despite Non-Uniform Invariant Measure

ISOMAP is a new nonlinear algorithm, which was recently developed in the field of data mining, to reduce the dimensionality of a data-set when a low-dimensional embedding manifold exists. In dynamical systems, when the attracting set is embedded in a lower dimensional manifold, we will show how ISOMAP can be useful to reduce the dimensionality of useful models of the system. The major problem of the issue however is that while, the major convergence theorem of the ISOMAP algorithm [2] assumes a uniform distribution of data, and dynamical systems tend to have nonuniform, and often even fractal invariant densities, we will show here how experimentally collected data can be properly handled and modeled in the intrinsic coordinates of the embedding manifold. In this paper, our proofs will be for dynamical systems with hyperbolic invariant sets, which we believe give some insight into how the same methods can be applied more widely, including to the Lorenz differential equation example given at the end.