Talk:Minimal dynamical systems
This is an excellent article and I highly recommend acceptance. It is thorough, informative, authoritative and well-organized.
The authors have already incorporated a set of very minor corrections and comments on the first half of the article, which I made a few weeks ago. I apologize for not finishing the job right away. I respect the authors' preference for the old-fashioned reviewing process; however, I find it much less time-consuming to make small changes directly in the text; as I understand it, the authors can see where the changes are made and accept or reject them. As I am over deadline and do not want to delay the article over small matters, I am accepting it immediately, without waiting for revisions.
Comments of the second reviewer:
I agree that "this is an excellent article" and "it is thorough, informative, authoritative and well-organized". However, I have some minor comments about the exposition.
1) The motivation ("... it may first be useful to find its nontrivial closed subsystems and study ...") is not very convincing. Maybe it is better to say that minimal systems are natural generalizations of periodic orbits, and they are analogues of ergodic measures in topological dynamics?
2) The word "proper" is used throughout the article in the meaning "not equal to the whole space". This should be explained, since quite often the meaning of this word is "not equal to the whole space and not empty".
3) In the section "Minimal systems - equivalent definitions", in the second paragraph, "set" should be "subset" (we do not want to speak of a "proper set").
4) In the section "Existence of minimal sets", in the second paragraph, it can be added that this follows immediately from the Zorn's Lemma.
5) At the end of the section "Minimality and syndetical recurrence", it is not good to mix the complex and polar notation. Either both formulas should be in the complex notation (then the modulus and argument can be used), or both should be in the polar coordinates (then \(X\) is given by \(0\leq r\leq 1\)).
6) Near the beginning of the section "Examples of minimal non-invertible maps", I would replace "endowed" by "together".
7) In the section "Spaces admitting minimal maps" it is written: "It is known (Church [Ch]) that any \(C^\omega\) (real analytic) monotone map on a compact (smooth) manifold is a diffeomorphism." It seems that the assumption that the map is not constant is missing.
8) In the last paragraph of the section "Spaces admitting minimal maps", the term "degenerate" should be explained (or replaced by its description).
9) In the section "Topological transformation groups and minimality", the term "discrete flow" is an oxymoron ( http://en.wikipedia.org/wiki/Oxymoron ). Can it be replaced by something else?
10) In the last paragraph of that section, delete "it holds that".
After the revision it is excellent.