Talk:Takens' theorem

From Wikipedia, the free encyclopedia

This article is within the scope of the following WikiProjects:

WikiProject Physics Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics. ??? This article has not yet received a rating on the assessment scale. [FAQ] ??? This article has not yet received an importance rating within physics. Help with this template Please rate this article, and then leave comments to explain the ratings and/or to identify its strengths and weaknesses. WikiProject Systems     (Rated Start-Class) Systems science Portal This article is within the scope of WikiProject Systems, which collaborates on articles related to Systems science. Systems rating: Start Class Mid importance  Field: Dynamical systems Please update this rating as the article progresses, or if the rating is inaccurate. Please also add comments to suggest improvements to the article.

[edit] Confusing

I find this article confusing because of the way it talks about Hausdorff dimension and Whitney's theorem. In particular, it says:

Assume that the dynamics f has an invariant manifold A with Hausdorff dimension d_A (it could be a strange attractor). Using Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with k > 2 d_A.

The Whitney's embedding theorem I know (and the article in Wikipedia) does not talk about Hausdorff dimension. The version of Takens theorem I had heard also does not deal with Hausdorff dimension, just plain old topological dimension. I have a feeling that this article might not be quite right--or is it some generalization I hadn't heard before?? does anyone know? Would it be better to have the usual statement of Takens' theorem here? --Experiment123 18:58, 14 March 2006 (UTC)

I wrote the article from memory while I was writing something on Takens. What I wrote down is most likely tainted by Sauer and friends (I remember wondering about the one that is in the original k > dim_H(A×A) + 1). Upon re-reading, it should be "using ideas from Whitney's theorem" and "manifold" should be set. — XaosBits 01:51, 15 March 2006 (UTC)

Cool! Looks good! --Experiment123 02:32, 15 March 2006 (UTC)

Fixed the Hausdorff dimension (that was wrong). The Sauer-Yorke-Casdagli version works only with box counting dimensions and not Hausdorff. Added the original references. I also feel that Takens' theorem should redirect here and the page should be moved to something more generic. — XaosBits 16:29, 15 March 2006 (UTC)