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 subsets $A$ of Euclidean space $R^k$. If $n$ is an integer larger than twice the box-counting dimension of $A$, then almost every map from $R^k$ to $R^n$, in the sense of prevalence, is one-to-one on $A$, and moreover is an embedding on smooth manifolds contained within $A$. If $A$ is a chaotic attractor of a typical dynamical system, then the same is true for almost every delay-coordinate map $R^k$ to $R^n$. These results are extended in two other directions. Similar results are proved in the $m$ general case of reconstructions which use moving averages of delay coordinates. Secondly, information is given on the self-intersection set that exists when $n$ is less than or equal to twice the box-counting dimension of $A$.