Talk:Takens's theorem
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Confusing
editI 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 dA (it could be a strange attractor). Using Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with k > 2 dA.
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 > dimH(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)
Simplified, slightly inaccurate version
editThis section seems oddly informal by Wikipedia standards. It would help to state what aspects of the description are simplified and inaccurate, and to note when these differences from the formal description are relevant. — Preceding unsigned comment added by 99.250.166.255 (talk) 05:36, 7 April 2017 (UTC)
Typo in the title?
editShouldn't the title be "Takens' theorem" (instead "Takens's theorem") ?
Inconsistency in the informal section?
editThe dimension "d" as defined in the informal section appears to be the same as "nu" in the previous section. It may be easily confused with d_A in the previous section. Also, the informal section states that k is at most 2d+1, but nowhere in the previous section does it state that k < 2nu+1 or k < 2d_A+1. In fact, the previous section states that k > 2d_a, which is a lower bound, not an upper bound on k, so it is very unclear how the informal section relates to the previous section. The readability of the article can be improved by making it clearer how the two sections are related. — Preceding unsigned comment added by 192.16.204.80 (talk) 21:57, 4 March 2022 (UTC)