At some point I've got to bug you about the notion that I've had chicken scratching at the back of my mind but I really don't have the appropriate understanding of the industrial machinery of algebraic geometry to make sense of -- that of thinking of strange attractors as sorts of algebraic varieties (well, the trivial case of a fixed point is a gimme, but I don't know where to go from there).
I gather that this has become the dreaded "some point". Confound it, you made me have to think, and you're exposing my experience level to the public even further.
I should ask if you mean a specific class of strange attractors, or precisely what you mean by "sorts of". In my limited exposure i've only come across SAs as embeddings of R in some Rn, and by virtues of nonrepetition and compactness of some neighborhood in Rn containing the image, they must have (Euclidean) limit points not in the image. Such things cannot be varieties, which as Zariski-closed sets are necessarily Euclidean-closed. (If you don't know the Zariski, it's enough to think of closed sets as zero sets of collections of equations and to note that the topology is strictly coarser than the Euclidean.)
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I should ask if you mean a specific class of strange attractors, or precisely what you mean by "sorts of". In my limited exposure i've only come across SAs as embeddings of R in some Rn, and by virtues of nonrepetition and compactness of some neighborhood in Rn containing the image, they must have (Euclidean) limit points not in the image. Such things cannot be varieties, which as Zariski-closed sets are necessarily Euclidean-closed. (If you don't know the Zariski, it's enough to think of closed sets as zero sets of collections of equations and to note that the topology is strictly coarser than the Euclidean.)
Is this along your desired lines?
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