Talk:Kneading theory

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    I made a couple of changes in the article, the way the author prefers. However, there are problems with the article that cannot be addressed easily that way.

    This is a very nice, technical accurate, and detailed account of the kneading theory in its Milnor-Thurston version (in fact, it is more detailed than most of other articles in Scholarpedia). However, I see a problem with this. Namely, there exists another approach to the kneading theory, which (in my experience) is more popular than the Milnor-Thurston one. I am not sure who introduced it, but it appears for instance in the book of P. Collet and J.-P. Eckmann "Iterated maps on the interval as dynamical systems" (Birkhauser, Boston, 1980). For instance, for unimodal maps, one uses symbols L and R instead of +1 and -1, and additionally there is a symbol C that stands for the critical (turning) point. The main difference is that kneading sequences or itineraries of the trajectories that fall into the critical point are truncated there, instead of being looked upon as one-sided limits. Renormalization corresponds to the *-product of kneading sequences. There are also some philosophical differences between the approaches. While the Milnor-Thurston approach is quite algebraic and leads to nice general theorems, the Collet-Eckmann approach is rather aimed at providing a useful language to work with piecewise monotone interval maps. For unimodal and more complicated cases (multimodal, piecewise continuous, tree maps, etc.) one tries to characterize which sets of sequences can appear as kneading ones, which have to appear in "full" families, or if we fix kneading sequences, what itineraries have to appear. Then in other papers those results are used, or sometimes no special results are necessary, just this is a convenient language.

    Therefore I think that the Collet-Eckmann approach should be also presented in the article. Moreover, it should not be just mentioned in the section "Other directions," but should be given a more prominent place.

    One additional minor remark: in Figure 2 the vertical axis is marked with the values of topological entropy 0.2, 0.3, etc. This may be meaningless to the reader, since various people use various bases of logarithms, and often even the base is not specified. The common way of thinking of the values of the entropy is as the logarithm of a given number, and this is how the vertical axis should be marked.

    I am grateful to the reviewer for his or her careful reading of the article, and for the specific changes made, which certainly improve it. Also for the proposed change to Figure 2, which I have implemented.

    Of course I am aware of the alternative approach which the reviewer describes. When planning this article I thought very carefully about which approach I should take, and which aspects of kneading theory I should emphasize: my aims were to avoid the article becoming too long (I failed in that) and to try to ensure that the flow of the article was coherent and linear. In the end I decided to describe kneading theory a la Milnor-Thurston, for exactly the reasons given by the reviewer: it is a clean and unified approach which yields relatively straightforward proofs of interesting general theorems.

    I agree that in doing this there is much that has been left out, as summarised by the reviewer: my intention in writing the first two bullet points of the "Other directions" section was to hint at this missing matter. However, I do not agree that it would improve the article to describe this in more detail: as I say, I considered this very carefully before writing the article and decided that it would make the article too long, and its structure over-complicated. Of course this is not to say that it wouldn't be valuable to have another Scholarpedia page treating this more combinatorial aspect of kneading theory.

    I understand this point of view, so let it stay as it is. However, I am afraid that there may be readers who come to this article after encountering the other approach. They can be greatly confused. Therefore I propose to add a very short (5 lines) section about the existence of another notation, one line in the introduction referring to this section, and one reference (look at the changes in the article). This will not add much to the length and will not change much the structure of the article. What do you think?

    Thank you for your understanding of my approach. I am very happy with the additions you made, although I removed the word "original" in the introduction, since as I understand it the RLC notation predates Milnor and Thurston's work.

    Thanks also for correcting the typos.

    /Review by Charles Tresser

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