June 7, 2007
In terms of technical information, the article is fine (in fact, I would defer to Misi- urewicz’s much greater expertise in these matters). My comments concern only readability and grace. The main issue is the vexing the. My only semi-substantive comment is the last one.
• applied for should read applied to
• in dimension 1 combinatorial should read in dimension 1 the combinatorial
• the Braid Theory should read Braid Theory Interval
• the following Sharkovsky’s ordering should read the fol lowing Sharkovsky ordering or just Sharkovsky ordering (somehow, the article “the” collides with the possessive “Sharkovsky’s”) This occurs again later.
• Note that Sharkovsky’s theorem also works for continuous maps of the whole real line (in fact, if I recall, that is the context in which he formulated his theorem).
• Minimal possible should read The minimal possible
• with a cycle... should read which has a cycle... (to my ear, the repetition of “with”’s in the current version is cacaphonic) • the most forms should read most forms
• (which additionally... should read (which also...
• has been identified should read have been identified
• or ∅ should read or it is empty
• Then periods are all denominators should read Then the periods are all the denominators
• Minimal entropy should read The minimal entropy Trees
• the Sharkovsky’s should read Sharkovsky’s or “the Sharkovsky” (see under Interval above) Graphs
• only several should read only a few Dimension 2
I don’t understand the comment:
Similar questions can be asked as for the interval maps, although usually they are much harder.
I think the reference to Bestvina-Handel as giving a way of deciding forcing is misleading: as I recall, there is no procedure there for deciding whether a given type forces another one. Handel had a short paper in which he did something like that for period 2 or 3 only, and otherwise the main work of this sort that I know is the work of Carvalho and Hall, which I think should be referenced, as well.
This is an excellent and accurate survey of an interesting and important area of Dynamical Systems written by one of the international leaders in the field.
I think it would be stronger as an encyclopedia article if it moved beyond the work of the author and his collaborators and covered more of the material that is considered standard in one-dimensional combinatorial dynamics. For example, the topics covered in the section of the book by de Melo and Van Strien with that title, namely, kneading theory and renormalization among others.
If the author wants to leave the article as is I suggest it be renamed "Forcing relations in dynamics".
The name "Combinatorial Dynamics" does not apply to the material covered in Chapter II of the book by de Melo and van Strien. Even the AMS Mathematics Subject Classification section 37E15, Combinatorial dynamics, includes an explanation "types of periodic orbits." Kneading Theory and renormalization should be covered by separate entries in Scholarpedia.
To avoid further confusion, I added a sentence about the book of de Melo and van Strien.