Author's Second Response to Reviews (07/21/07)
All additional suggestions by the referees (including the new referee C), are met besides the following ones. I believe that "Dynamical Billiards" is a natural name for a WEB real life encyclopedia (otherwise real billiard will pop up when one clicks "billiards"). The paper Bunimovich, DelMagno (2006) deals with semi-focusing components rather than with focusing ones. It was supposed that some links to the existing on the WEB animations of dynamical billiards will be added. This will take care of the Referee A comment that the illustrations are poor.
Author's First Response to Reviews
First, I'd like to thank both referees for their comments and suggestions. I have followed almost all of them with a very few exceptions.
The Reviewer A has two general remarks and I do disagree with both of them. I believe that both these remarks are based on quite different view of what Scholarpedia is (or is supposed to be). First of all it is Encyclopedia. It means that it is allowed (and should be encouraged) to use "technical terms" which are introduced in details in other articles of Scholarpedia. It is not a popular review article. Moreover, I had (and still have) a feeling that the articles should be kept within four pages or at least close enough to this bound. Secondly, it is a scientific Encyclopedia for scientists. Observe also that it is also very focused: it is not an Encyclopedia of Sciences, not a Mathematical Encyclopedia. It is focused on neuroscience and general ideas, notions and tools of the theory of dynamical systems. The emphasis here is on "general," rather than on very specific, although nice and difficult, mathematical questions.
I did include hard balls Boltzmann gas as one of the major motivations to study billiards as well as references to Boltzmann-Sinai ergodic hypothesis. It is a standard way to start a review paper on billiards. In fact, I thought that Scholarpedia will include a separate article on foundations of statistical mechanics which motivated the creation and development of Ergodic Theory. I still think that it'd be a good idea to include such article.
Observe that "Dynamical Billiards" appear in Scholarpedia in "Ergodic Theory" and "Hyperbolic Dynamics." Therefore it is clear and natural that the emphasis should be on typical billiards and their generic behavior. Typical billiards have chaotic (hyperbolic) components of positive measure. Thus the mechanisms of chaos should be the main points of discussion. So it necessarily should be discussed both mechanisms of chaos and natural obstacles like astigmatism to their existence. Therefore I completely disagree with the Reviewer A comment about "rather special multidimensional constructions." Actually I am very puzzled by this comment because in fact there is (and there was) no word in my article about these constructions besides that focusing components cannot be large in d > 2 because of astigmatism.
I admire the results, techniques and the area of polygonal billiards. In fact, I feel that mathematics there is more beautiful than in hyperbolic billiards. However, polygonal billiards, especially in rational polygons are extremely nontypical billiards. Their studies explore very sophisticated mathematics and they are concerned with a very fine and very specific structure of orbits. There is no room here to properly introduce and define the needed notions. Observe that interval exchange transformations, which are more general are not even discussed in Scholarpedia. Nevertheless, I did include a section on billiards in polygons and polyhedra. I didn't include outer (or external) billiards because they are not billiards. Those are very nice dynamical systems which have no relations to billiards in Euclidean spaces which are the only one considered in this article.
I consider the first book on billiards by S. Tabachnikov to be the best existing introduction to the theory. In fact it was in my list of references but was omitted (together with Lazutkin, 1973) in typing. I included all the references suggested by both referees besides two. The paper by Sinai is already outdated and covered better by the other references. I didn't know about Katok's lecture. However after reading I didn't include it because of some factual and historical inaccuracies. I didn't include numerous references to the studies of rational polygonal billiards but put a reference to a recent review. Also, I didn't include a mirror formula (although it is my favorite tool in billiards) because it is a technical tool and it'd need to include explanations why and where it is useful. Otherwise, all the suggestions and comments are taken into account. Finally, I'd like again to thank the reviewers for their thoughtful input.
As I understand this article should be written for general public, not for experts. In my opinion it should introduce the topic in common (simple) terms, provide a few most basic definitions and facts, relate the topic in question to other areas in sciences, and supply references for further reading.
In this respect, I have two general remarks:
First, the author often uses special technical terms without defining or describing them, for example "configuration space", "phase space", "orbit", "singularity", etc. I suggest the author attempts to use simpler language to improve readability. Special terms should be defined or described.
Editor-in-chief has included some links to articles describing these notions.
Second, the presentation is heavily biased and focused on the author's own contribution. His focusing billiards are discussed in detail, including his rather special multidimensional constructions. On the other hand, mechanical models of hard colliding balls (hard ball gases), which originally motivated Sinai to study chaotic billiards and which still supply the most challenging billiard-related problems, are not even mentioned. The author could have mentioned intensive studies of periodic orbits (going back to Birkhoff), dual billiards, and profound recent results by Burago et al on counting reflections near corner points (that is related to hard balls again). I also suggest to write a separate (larger) section on polygonal billiards, including those with rational angles, rather than casually mentioning them in the middle of a section on focusing billiards.
Including outstanding open problem besides Birkhoff's conjecture (say, ergodicity of hard ball gases, existence of periodic orbits in any polygon, etc.) might be a good idea, too.
A few other remarks:
1. References should include one of the best books around on the subject: "Billiards" by S. Tabachnikov (Panorama and Syntheses, 1995).
2. The paper (Lasutkin, 1971) is mentioned in the text but missing from the reference list.
3. "American Mathematical Society Press" is an inaccurate description of the publishing house. For example: "Tabachnikov... AMS, Student Mathematical Library, Vol. 30". Same goes to the books by Kozlov & Treshchev and by Chernov & Markarian. (The book by Birkhoff is referred to correctly.)
I approve the article.
My general impression of the article is positive. What follows are a several suggestions to the author.
1. I would mention the "mirror equation" of geometric optics; this equation plays a major role in the study of billiards.
2. I would mention Mather's theorem on non-existence of caustics (a very natural place for that is when the existence of caustics is discussed).
3. I believe, billiards in polygons deserve a separate new section: there is a very substantial activity on rational polygons (Veech, Masur, Smillie, Eskin, Zorich, just to mention a few names).
4. I would mention Birkhoff's periodic trajectories and, possibly, recent multi-dimensional generalizations (Farber-Tabachnikov).
5. I would mention recent progress toward proof of the Boltzmann-Sinai ergodic hypothesis.
6. I would mention recent results on the number of collisions in systems of elastic balls (Burago and his coworkers).
7. I'd mention various mechanical systems with elastic collisions that motivate the study of billiards.
(i) l. 7 of "General Properties": "... is equal TO zero..."
(ii) Section "Integrable Billiards": Berger's theorem holds in all dimensions not less than three; one should refer to a different, longer, paper of his on the subject (Bull. Soc. Math. France 123 (1995), 107--116.)
(iii) Section "Chaotic Billiards" with one line content makes a strange impression; delete it.
(iv) In the references, I'd add Ya. Sinai's ICM 1990 talk "Hyperbolic billiards"; S. Tabachnikov's book "Billiards" (SMF "Panoramas et Syntheses", No. 1, 1995); references to the recent "Handbook of Dynamical Systems"; and a survey by A. Katok "Billiard table as a playground for a mathematician" in Surveys in modern mathematics, 216--242, Cambridge Univ. Press, 2005.
I am satisfied with the changes made and endorse the paper. Two items for the author to correct:
1). 1st sentence of "Billiards in polygons..." should read: "Let ... HAVE a flat ..."
2). Birkhoff's theorem asserts that for every $n$ and every $r\leq n/2$, coprime with $n$, there exist at least two $n$-periodic billiard trajectories with rotation number $r$. The current formulation is too weak.
I completely agree with the remarks made by the Reviewers A and B. I add the following obvservations:
0) I do not like the title of the article. All billiards are dynamical.
1) Dispersing billiards. It is not true that "all these properties (ergodicity, ..., exponential decay of correlations) are ensured by the mechanism of dispersing." Dispersing billiards with cusps have polynomial decay of correlations (Chernov- Markarian).
2) Focusing billiards. I do not understand the exact meaning of the following sentence: "The mechanism of defocusing... average (over the invariant measure)time of convergence."
3) Chaos and astigmatism. The mechanism of defocusing works as in demension 2, in semi-focusing billiards with cylindrical components (Bunimovich-DelMagno).
4) There are a lot of unjustified rigid assertions: "most visual", "All these properties", "there is no other mechanism of chaos", "The most general class", "The only visual and rigorously".
I suggest that the author rewrite the article, following the recommendations of all three Reviewers.
First of all (this is not related to the article), the refereeing system is confusing and I'm not sure that I'm typing my comments in the right place. I hope the author will see them anyway.
The author made many improvements, and the article is much better now. In my view, it is still written for a narrow circle of experts, rather than for a broader science community. But the author argues (in his response to the reviewers) that his article's style is in line with others in Scholarpedia, so I drop my criticism on that.
I also feel that the author could have provided better illustrations: he states that "billiards form the most visual class of Hamiltonian systems" (by the way, such categorical claims in the article appear to be too rigid), but the visual images he provides are rather poorly drawn and simplistic. One could use colors, moving visual elements, or make an interactive picture by Java code.
Lastly I have a few minor suggestions:
"reflections from" --> "reftections off (or at)" "free passes" --> "free paths" "Let ... has .." --> "Let ... have ..." "a typical billiard in a polygon" --> "billiard in a typical polygon"
elliptical and hyperbolic caustics are NOT separated, they intersect each other all over in the ellipse; it is the corresponding trajectories in the phase space that are separated.
Sinai billiards have exponential decay of correlations only if the corner points are not cusps (this condition is missing in the article).