Talk:Maps with vanishing denominators

Reviewer A

This entry on vanishing denominators should be enlarged to include more references to work other than that by the authors. The reviewer also thinks that the title "vanishing denominators" is rather obscure and could be changed. [Editor: Perhaps the concept studied is "rational maps", since all of the maps refered to in the paper are ratios of polynomials. Would this be more appropriate?] The dynamical concepts captured by the term "vanishing denominators" have been studied in a more rigorous and complete way in a variety of papers of other authors (to which no citations are made). The entry uses some obscure terminology for standard concepts that should be improved. I elaborate with some more specific technical comments below.

Instead of the term "set of nondefinition," use the standard term "complement of the domain."

Several times, sets need to be enclosed in brackets.

Instead of the term "two-dimensional recurrence," use the standard term "forward orbit."

Instead of the term "non interrupted sequences," use the standard term "full forward orbit."

It is not always the case that $R^2 \setminus E$ has Lebesgue measure zero. Thus it should be stated as either the informal "in general" or the more formal "generically" (in some specific function space).

There is a reference to Fig1a which should refer to Fig1b.

The section "Case of a noninvertible map" is neither important nor relevant. It is just a review of the articles of the authors of the entry.

Figure 2 is too complicated to be helpful without a clear description. The description given is extremely technical without any clarity.

The section "Some dynamic properties of focal points" makes some interesting statements about invariant manifolds. However, these statements have been already discovered by others albeit without the use of the term "vanishing denominators." It is inappropriate to state these results without citing the main original contributors to their discovery.

The term "lobes" is a bad choice, since it has a standard use in the dynamical systems community.

The "knots" referred to at the end of the article have been seen in other contexts. See the previous comment about the omission of citations.

There are numerous language mistakes, but I do not enumerate them in this review.

Below is a list of some papers which include concepts mentioned in this entry.

L. Billings, J. H. Curry and E. Phipps, "Lyapunov Exponents, Singularities and a Riddling Bifurcation," Physical Review Letters, 79: 1018-1021, 1997.

L. Billings and J. H. Curry, "On Noninvertible Mappings of the Plane: Eruptions," CHAOS, 6:108-120, 1996.

C. E. Frouzakis, I. G. Kevrekidis, and B. B. Peckham, A route to computational chaos revisited: Noninvertibility and the breakup of an invariant circle, Phys. D, 177:101-121, 2003.

K. Josic and E. Sander. The Structure of Synchronization Sets for Noninvertible Systems . Chaos, 14(2):249-262, 2004.

Bernd Krauskopf, Hinke M. Osinga, and Bruce B. Peckham. Unfolding the Cusp-Cusp Bifurcation of Planar Endomorphisms. SIADS 6(2):403-440, 2007.

E. N. Lorenz, Computational chaos--a prelude to computational instability, Phys. D, 35:299-317, 1989.

V. Maistrenko, Yu. Maistrenko, and E. Mosekilde, Torus breakdown in noninvertible maps, Phys. Rev. E (3), 67 (2003), 046215.

C.-H. Nien, The dynamics of planar quadratic maps with nonempty bounded critical set, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8:95-105, 1998.

E. Sander Homoclinic tangles for noninvertible maps. Nonlinear Analysis, 41(1-2):259-276, 2000.

Authors Reply to Reviewer A

Parts in italics concern the authors answer. The other parts reproduce the reviewer text

Introduction

We are conscious that the subject, and the related terminology, is not so widely diffused as other topics already published in scholarpedia. Indeed, the dynamical problems related with such kind of maps is quite old, but the results and terminology introduced to explain such dynamic phenomena are very recent, hence not yet well established. We will easily follow the minor remarks of the referee in order to improve our paper; however, it is not clear to use that the reviewer has supplied any additional references on the subject of this article---the supplied references all seem to be about noninvertible maps, and not to maps with a vanishing denominator For example, the contents of the main page gives section 2 with the clear title "invertible maps", section 3 with the title "Noninvertible maps". It is the same for Figs. 2 and 5, the only ones for which the captions visibly indicate the noninvertible case. All the remaining parts and figures of the main page concern invertible maps, with sometime references to the noninvertible case. Ditto for the subpage contents with the sec. 4 title "Polynomial invertible map whose inverse has a focal point".

Reply to each point of the review

• This entry on vanishing denominators should be enlarged to include more references to work other than that by the authors. The reviewer also thinks that the title "vanishing denominators" is rather obscure and could be changed.

The fact that we (the authors) are citing only papers of their own group is due to the fact that the subject was treated only in those papers, at least from the generic point of view (a few applications appeared after, and maybe we can add them, however there is no space, and the applications published after use our results referring to our papers, without adding general results). To the references added by the referee we shall answer here below, anyhow, except the two papers by Billings, which we know because they motivated one of our papers (Bischi, Fournier, Gardini "Basin boundaries and focal points in a map coming from Bairstow's STRmethods" CHAOS, (1999), vol.9 n.2,)" not cited because it is a simple application of the theory), dealing with maps associated with the Bairstow’s methods, all the others concern non invertible maps (denoted below NIM) without a vanishing denominator, which have nothing in common with the subject of our article. But more, the paper by Billings and Curry is cited by us in the applications (not in the masterpage, but in the subpage, for the reason written above: in the two papers by Billing cited by the referee there is nothing related to focal points and prefocal sets, which is the key to understand the dynamics of such maps, neither called with different names. While, as we have shown in the paper Bischi-Fournier-Gardini, the dynamics observed by those authors can be EXPLAINED taking into account the subject of the present article.

It is difficult to invoke papers published after our basic ones Mira C. [1981]. (presented by René Thom), Bischi et al. [1999, 2000, 2003] for contesting our originality, all the more reason with papers (for most of them) quoting the authors results but dealing with NIM without denominator. Moreover in sec. 1 of "For authors" the editor asks "Keep references to the minimum, citing only major books and review articles." It is the reason why our article does not quote our papers written before 1999 (except Mira C. [1981] which from 2 examples describes focal points properties, and introduces the name. What is surprising is that except Billings, the referee's references concern NIM WITHOUT DENOMINATOR.

The term "vanishing denominators" is not obscure, but completely clear for all those dealing with rational maps which are not associated with the complex plane. Although an equivalent term is “maps not defined in the whole phase space” we prefer the above emphasis on the denominators.

• Furthermore, the dynamical concepts captured by the term "vanishing denominators" have been studied in a more rigorous and complete way in a variety of papers of other authors (to which no citations are made).

This affirmation is not correct: to our knowledge this subject in “iterated maps” (i.e. not from the simple geometric point of view) has been studied only in our papers (and several applications appeared after which refer to our papers). If the referee knows some other papers, please let us know the specific references.

It is worth noting that, according to the Scholarpedia recommendation, the article did not cite previous authors papers about the topic such as:

• • Bischi G.I. & Gardini L., «Focal points and basin fractalization in a class of triangular maps». Proceedings of European Conference on iteration theory, ECIT96, Urbino (8-14 Sept., 1996).

• • Mira C. "Some properties of two-dimensional nondefined maps in the whole plane". Proceedings of European Conference on iteration theory, ECIT96, Urbino (8-14 Sept., 1996).

• • Bischi G.I. & Naimzada A., «A coweb model with learning effects». Atti XiX Convegno AMASES, Cacucci Editore, Bari, 1995, 162-177.

• • Bischi G.I. & Naimzada A., «Global analysis of a nonlinear model with learning». Economics Notes, vol. 27 (3), 143-174. 1997.

• • G.I. Bischi and L. Gardini, «Basin fractalization due to focal points in a class of triangular maps», International Journal of Bifurcation and Chaos, vol.7 (7), pp. 1555-1577 (1997).

In the "dynamical systems" framework, or even (in spite of a nonstrict relation) in other fields (such as the literature on algebraic geometry), if the reviewer finds previous references about the theory before 1996, and about examples before the Mira [1981] paper, this will be considered by us as a precious help, because in spite of our negative researches, it may be possible that such works exist.

• The entry uses obscure terminology for standard concepts.

"We believe our terminology is well-defined. If there are “obscure terms” in place of a standard concepts, (apart from the specific comments of the referee), we ask the referee to indicate them in more detail.

• The fact that most of the terminology in this article is new terms for standard concepts - along with the comment at the end of the article that there is significant overlap between the ideas here and the literature on algebraic geometry - indicate that the use of standard terminology would alleviate the need for this article.

The use of a different terminology might be done perhaps in the future by someone else (as we are not interested in this). And even if somebody may have a different terminology in the geometric context, maybe this was done-given after our works. Anyhow, it seems to us more correct to emphasize that the role in iterated maps (i.e. in dynamics) is different, so the use of different terminologies with respect to those used in the geometric case is more suitable (for example one iteration of a map cannot reveal homoclinic contacts and other bifurcations)

• The concepts described here could be added with appropriate citations within the definition of noninvertible maps (Mira is already an author for this article).

We tried to do this but the subject is too long to be explained in a few sentences. A whole paper is necessary only to clarify some basic properties. And we hope that the readers can apply the theory to their maps. A link with maps (invertible or not) without denominator exits, when the inverse map has a denominator inducing specific properties described in sec.5 of our article main page, and the polynomial map of example 4 of the subpage (sec.4). It is impossible to incorporate the present article to the article "Noninvertible maps" because it supposes the whole knowledge of the article. It is why "Noninvertible maps" has a link with the article for "knots". Moreover the present article concerns not only NONINVERTIBLE MAPS but also INVERTIBLE MAPS, and so cannot be included in the article "Noninvertible maps".

• We add some more specific technical comments below.

• • Instead of the term "set of nondefinition," use the standard term "complement of the domain."

No, "complement of the domain" is not a suitable term. For example a map can have in its definition log(x+y) or something else, for which a “natural domain of definition exists”, but we are not interested in such kinds of domain and to its complement. And if log(x+y) is in the denominator, the “set of non definition for the map” is (x+y)<=0 and (x+y)=1, but the set of nondefinition for the rational map (as we have defined) is only the set in which the denominator becomes zero, i.e. (x+y)=1.

• • Several times, sets need to be enclosed in brackets.

We thank the referee for this remark. Indeed it was a technical problem, and we have seen that the brackets have disappeared in two definitions of \(\delta_{s}\), and \(J_{0}\).

• • Instead of the term "two-dimensional recurrence," use the standard term "forward orbit."

No. An “iterated map” is not an “orbit”. The term "Recurrence" is convenient to denote the repeated application of T, widely used since long time in the literature on Iteration Theory (see e.g. the book by Montel 1957, quoted in Abraham and Ueda, ”The chaos avant-garde” World Scientific Series on Nonlinear Science Series A vol. 39, p. 95-198 ). We do not understand why "forward orbit” would be the standard term. Moreover it is not suitable with respect to our works, because we need to emphasize that a recurrence may be non uniquely invertible, and the term “forward orbit” is really not standard in this context. May be “map” is the standard term! (but perhaps this depends on the different Journals!)

• • Instead of the term "non interrupted sequences," use the standard term "full forward orbit."

This suggestion may be accepted.

• • It is not always the case that R^2 has Lebesgue measure zero. Thus it should be stated as either the informal "in general" or the more formal "generically" (in some specific function space).

No, this comment is wrong. We assume that the set of non definition ( ) is a single curve or the union of smooth curves of the plane, which is a set of zero Lebesgue measure in R2. Thus all the possible preimages cannot give a set of higher dimension (we DO NOT take the closure of this set).

• • There is a reference to Fig1a which should refer to Fig1b.

No, the reference was correct.

• • The section "Case of a noninvertible map" is neither important nor relevant. It is just a review of the articles of the authors of the entry.

We disagree with this comment. The notions of importance and relevancy are relative, depending on the mathematical domain. And in the context of the paper the difference is important!

• • Figure 2 is too complicated to be helpful without a clear description. The description given is extremely technical without any clarity.

We thank the referee, may be this description may be improved

• • The section "Some dynamic properties of focal points" makes some interesting statements about invariant manifolds. However, these statements have been already discovered by others albeit without the use of the term "vanishing denominators." It is inappropriate to state these results without citing the main original contributors to their discovery.

To our knowledge, the results described in our papers related to rational maps have not been published elsewhere by other authors. The referee is encouraged to give specific references.

• • The term "lobes" is a bad choice, since it has a standard use in the dynamical systems community.

The use of "lobes" in dynamical systems is associated with the curves of stable and unstable manifolds when homoclinic points are formed, and such regions are relevant mainly for models in continuous time. But this is a completely different context. And there is no risk of misunderstanding. About what is called "Lobe dynamics" this concerns another field, the fluid mechanics ones.

• • The "knots" referred to at the end of the article have been seen in other contexts. See the previous comment about the omission of citations.

This is wrong, the references given below, except Billings, do not deal with maps with vanishing denominator. Moreover the most part of the papers, claimed as omitted, quote the Gardini, Gumowski, Mira papers on noninvertible maps WITHOUT DENOMINATOR, so without any relation with our present article.

• • There are numerous language mistakes, but I do not enumerate them in this review.

This is highly regrettable, such data would have been very useful for the article improvement .

Below is a list of some papers which include concepts mentioned in this entry.

Analysis of each of the papers cited

Except the two Billings papers (that we knew, one of them being quoted in the subpage article as application to a particular map), each of the other papers mentioned by the reviewer, only concern maps WITHOUT DENOMINATOR, and thus are not related to our present work. Several papers cited by the referee (for example the papers Frouzakis et al, Krauskopf et al., Maistrenko et al., Nien) had been eariler studied by us in our old articles, or in the chapter 6 of the book published in 1996 by C.Mira, L.Gardini, A. Barugola & J.C. Cathala "Chaotic dynamics in two-dimensional noninvertible maps" (World Scientific Series on Nonlinear Science), cited by these authors. The referee can see in this book that figs. 6.1, 6.19, 6.20, 6.21, 6.39, 6.40, 6.49, 6.52, 6.57, also present a kind of "vague similitude" with "lobes" and "knots"."'

• L. Billings, J. H. Curry and E. Phipps, "Lyapunov Exponents, Singularities and a Riddling Bifurcation," Physical Review Letters, 79: 1018-1021, 1997.

• L. Billings and J. H. Curry, "On Noninvertible Mappings of the Plane: Eruptions," CHAOS, 6:108-120, 1996.

This reference, being an application of numerical iterative methods (see Billings and Curry, [1996], is given in the subpage.

• C. E. Frouzakis, I. G. Kevrekidis, and B. B. Peckham, A route to computational chaos revisited: Noninvertibility and the breakup of an invariant circle, Phys. D, 177:101-121, 2003.

This paper deals only with noninvertible maps without denominator. For critical sets we quote Gumowski, Mira, Gardini, our 1996 book "Chaotic dynamics in two-dimensional noninvertible maps" (World Scientific Series on Nonlinear Science, Series A Vol. 20). In this paper the authors have used notions already published in a common paper: C.F. Frouzakis, L. Gardini, Y.G. Kevrekidis, G. Millerioux, C. Mira, "On some properties of invariant sets of two-dimensional noninvertible maps". Int. Journal of Bifurcation and Chaos 7(6), 1997, pp. 1167-1194, and also in the chapter 6 of our 1996 book. Nothing is related with denominators! Due to a kind of vague outline similitude in the figures, it seems that the referee mixes up "loops" due to self-intersections of the unstable set of a saddle in a NIM (WITHOUT DENOMINATOR), or due to homoclinic intersections in such a NIM (see figs. 6.1, 6.19, 6.20, 6.21, 6.39, 6.40, 6.49, 6.52, 6.57 chapter 6 of the 1996 book) , and the “lobes” issuing from focal points created by maps with vanishing denominators.

In the Frouzakis et al. paper, see equation (1) which is a polynomial noninvertible map (thus WITHOUT DENOMINATOR), figs. 9, 10 , 11, 12 related to self intersections with loops of the unstable set of a saddle (no relation with "lobes" and "knots" of our article) and quotations [1] [11] [14] [18] [23] [24] of our papers.

• K. Josic and E. Sander. The Structure of Synchronization Sets for Noninvertible Systems . Chaos, 14(2):249-262, 2004.

We have not found this paper in internet but, being related to synchronization problems, it is likely that it deals with NIM without denominator.

• Bernd Krauskopf, Hinke M. Osinga, and Bruce B. Peckham. Unfolding the Cusp-Cusp Bifurcation of Planar Endomorphisms. SIADS 6(2):403-440, 2007.

This paper deals only with noninvertible maps without denominator and about critical sets quote Gumowski, Mira, Gardini, our 1997 book, and others from Toulouse.

• E. N. Lorenz, Computational chaos--a prelude to computational instability, Phys. D, 35:299-317, 1989.

This paper motivated our common work: C.F. Frouzakis, L. Gardini, Y.G. Kevrekidis, G. Millerioux, C. Mira, "On some properties of invariant sets of two-dimensional noninvertible maps". Int. Journal of Bifurcation and Chaos 7(6), 1997, pp. 1167-1194. Nothing is related with DENOMINATORS!

• V. Maistrenko, Yu. Maistrenko, and E. Mosekilde, Torus breakdown in noninvertible maps, Phys. Rev. E (3), 67 (2003), 046215.

The subject of the paper does not deal with maps with denominator. In it, the authors show that the formation of “loops” is due to a particular situation of eigenvectors on the critical line, which is correct, and already reported in the text (not as a proposition) in our paper C.F. Frouzakis, L. Gardini, Y.G. Kevrekidis, G. Millerioux, C. Mira, "On some properties of invariant sets of two-dimensional noninvertible maps". Int. Journal of Bifurcation and Chaos 7(6), 1997, pp. 1167-1194.

• C.-H. Nien, The dynamics of planar quadratic maps with nonempty bounded critical set, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8:95-105, 1998.

The paper deals with planar quadratic polynomial maps, so without denominator. See the references of the paper, where 8 of them, with respect to the 18 quoted by this author, concern the authors of the Scholarpedia article.

• E. Sander Homoclinic tangles for noninvertible maps. Nonlinear Analysis, 41(1-2):259-276, 2000.

This paper also does not deal with focal points and maps with a vanishing denominator.

We would appreciate it if the reveiewer could give us additional references that refer to maps with vanishing denominator.

Suggestions of the review taken into account

We thanks the reviewer for his suggestions.

- References are added in the subpage (study of particular maps) to answer to the remark of non quotation of other authors. It is worth noting that the main page is only devoted to a theory of maps with vanishing denominators. If in the future previous references of other authors about this theory are found, they will be added. Now the beginning of the section "References" of the main page explains the limited citations of this page. It is important to respect the article structure (main page for theory, subpage for study of particular maps).

- The Figs. 1 and 2 captions are now completed, in particular the extended Fig. 2 one, which might be understandable in itself.

- "Non interrupted sequences" has been corrected, and also brackets have been put for the definitions of \(delta\ _{s}\), and \(J_{0}\).

- About "lobes" a parenthetical remark was put: (this usage of "lobe" is distinct from that used in transport theory).

Suggestions of the review not taken into account

- We think that, from the precise sec. 2.3 analysis, now the reviewer has clearly seen that his papers citations are not related to our article. But if he finds publications about the theory before our papers quoted in the main page, and examples before Mira [1981], this contribution will be very appreciated by the authors.

- About changing "Maps with vanishing denominators" into "Rational Maps" there is a big difficulty. Indeed the main page also concerns other function types in a denominator, such as: piecewise smooth (piecewise linear for example) functions, log(x+y), the concepts of focal point and prefocal set remaining the same.

For the other points see sec. 2.2 of our reply.

Reviewer B

I think the authors did a great job in describing the key sets that control the dynamics in maps with vanishing denominators. While I am comfortable with the mathematical validity of the page, I have the following minor editing suggestions:

First, I agree with the first reviewer that the title could be refined. My suggestion is "Vanishing Denominators in Rational Maps." I think that this would be an interesting result in a search for information about the dynamics of rational maps.

First paragraph:

• Needs a comma: At a focal point, one of the components has the form…

• Change wording: Roughly speaking, a prefocal curve is a set of points for which there exists at least one inverse set that maps (or "focalizes") the whole set into a single point, the focal point.

In the “Definitions and basic properties” section

• Needs a comma: Whenever you use the phrase “In this case,”

• I’m not sure what the authors are specifically describing when they use the term “The two-dimensional recurrence”. Recurrence usually refers to a refined algebraic relationship between iterates, which I do not see. I think that the authors are describing the set of initial conditions for well-defined, full forward orbits under the map and it should be plainly stated.

• You might want to quantify that the initial discussion of a simple focal point (above the focal point definition) is the case when gamma is nontangent to the set of nondefinition, delta. To be specific, I suggest that you add a comment that the tangent case is discussed in more detail below.

• Change wording of “and this implies that its Jacobian cannot be different from zero in the points” to “and this implies that its Jacobian cannot be nonzero in the points”.

In the “Case of an invertible map” section

• Change wording of “cannot to be strictly considered as an inverse” to “cannot be strictly considered as an inverse”.

In the “The focal points belong to the closure of the set of merging preimages…” section

• I do not find Figure 2 clearly illustrative of this case. Maybe it should be removed, along with the associated text that describes it. The section could rely upon a more clear discussion in the first paragraph and just direct the reader to Example 2 of the Subpage.

• Does an example in this case depend on the LC having two branches (hyperbola)? It seems that an important bifurcation occurs as those two branches converge, but the discussion only briefly touched on that fact in the example.

In the “Further remarks” section

• Change wording of “The theory of focal points and prefocal curves is also useful to understand some properties of maps” to “The theory of focal points and prefocal curves is also useful in understanding some properties of maps”

Reply to reviewer B

First the authors thank the reviewer for the time he devotes to the article improvement. They are aware of the amount of work that this task implies.

The corrections proposed were made, except the following points:

First, I agree with the first reviewer that the title could be refined. My suggestion is "Vanishing Denominators in Rational Maps." I think that this would be an interesting result in a search for information about the dynamics of rational maps.

About an eventual change of the title "Maps with vanishing denominators" into "Rational Maps", this would be a limitation of the topic to a restricted class of maps. Indeed in mathematics a rational function is any function which can be written as the ratio of two polynomial functions. Section 1 of the main page says that F, N, D are continuously differentiable functions. So for example one can consider in a denominator log(x+y). With complementary precautions it would be even possible to widen the hypotheses, for example with piecewise smooth (piecewise linear for example) functions, for which the concepts of focal point and prefocal set remain the same.

A compromise might be in adopting the title "Vanishing Denominators in Fractional Maps". The title is a little longer, and the Editor-in-Chief prefers short titles, but this might be possible.

• I do not find Figure 2 clearly illustrative of this case. Maybe it should be removed, along with the associated text that describes it. The section could rely upon a more clear discussion in the first paragraph and just direct the reader to Example 2 of the Subpage.

In order to clarify the figure 2 situation, its caption has been strongly enlarged by explaining the figure symbols. This figure is an important element for the article, it is not describing an exception but the general situation (supported by the example in the subpage).

• Does an example in this case depend on the LC having two branches (hyperbola)? It seems that an important bifurcation occurs as those two branches converge, but the discussion only briefly touched on that fact in the example.

The "bifurcation cases" are not described in this generic article which is an encyclopedia entry. For these cases a reference is made to Bischi, G.I., L. Gardini and C. Mira [2005]). "Plane Maps with Denominator. Part III: Non simple focal points and related bifurcations", International Journal of Bifurcation and Chaos, 15(2), 451-496.

Other point: a reference to the Harris' book (1992) dealing with algebraic geometry, published after Mira (1981), has been added in the main page.