Embedology

Mathematical formulations of the embedding methods commonly used for the reconstruction of attractors from data series are discussed. Embedding theorems, based on previous work by H. Whitney and F. Takens, are established for compact subsetsA of Euclidean space R^k. Ifn is an integer larger than twice the box-counting dimension ofA, then almost every map fromR ^k toR ⁿ , in the sense of prevalence, is one-to-one onA, and moreover is an embedding on smooth manifolds contained withinA. IfA is a chaotic attractor of a typical dynamical system, then the same is true for almost everydelay-coordinate map fromR ^k toR ⁿ . These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which use moving averages of delay coordinates. Second, information is given on the self-intersection set that exists whenn is less than or equal to twice the box-counting dimension ofA.