Talk:Kneading theory/Review by Charles Tresser

General comments

Context independent general remarks

This article, "Kneading Theory"" by Toby Hall is very well written and technically accurate, but in fact as a monograph on a small sub-subject of one dimensional dynamics, which is itself an important but quite limited subject in classical dynamical systems theory (as opposed, e.g, to dynamics when one takes relativistic and quantum mechanical effects) with an intended readership of well trained mathematicians. One would rather expect an encyclopedia style article, where the main results would be cited but no proof given, except perhaps if some special technique or idea of proof needs to be singled out, or a particularly short proof of a particularly important result, even in a partial form, can be provided. One would also expect that most readers come out with a clear reason of why the subject of the article deserves to be part of the entries of the encyclopedia (especially now when the list of planed articles is still rather short). To the defense of the author, the guide-lines (as far as this reviewer understands) where of the form: "do like Milnor in his (masterful) "attractor" article", but that is a bit like asking to be like God. The article in its present form does not at all take into account that the majority of people interested in (classical) dynamical systems theory are scientists (physicists, chemists, biologists,.etc.) and engineers and not mathematicians: some of these critics will resonate in the next subsection of this report, the "Context dependent remarks".

Context dependent general remarks

This masterfully written article (if this would be a monograph rather than an article in Scholarpedoia: see the "Context independent general remarks" above) omits to explore more carefully the pre-history of kneading theory: talking in particular of the 1973 work of Metropolis, Stein, Stein (and mentioning the earlier work of Myrberg), as well as the work of Derrida, Pomeau, and Gervois (preceded by the work of Mira and his school) would also have permitted to illustrate some applications of the kneading symbolics (all examples provided in the article are simple examples inside kneading theory or almost there: no perspective is provided in applications far enough from kneading theory, inside and out of mathematics). Perhaps even more important, it would have been nice to have a paragraph explaining how kneading theory is different from more main-stream aspects of symbolic dynamics and point out that while kneading invariants are attached to maps and permit us to know a lot about the global orbit structure of the maps (and the full list of orbits under some conditions), their evolution with parameter permits to describe (the main aspects of) the bifurcation structure of parameterized families (in general with as many parameters as turning points except if some condition such as a symmetry reduces the number of parameters that are needed). Such aspects would probably be more talking to users of dynamics than a discussion of kneading determinant, formal power series, zeta functions, which are all important and very well treated by the author but should be detailed after more concrete aspects of the kneading sequences and their applications are mentioned. In fact, besides participating to the seminal Milnor-Thurston paper that illustrated the piecewise linear maps that model all maps up to semi-conjugacy to a map whose entropy is the constant absolute value of the slope, Milnor was also the senior author of a paper by Milnor and Tresser where kneading theory is revisited and the main model consists in Stunted Sawtooth Maps (using ideas going back to Chenciner-Gambaudo-Tresser for circle maps and Múbru for the bimodal maps) such that the full family of such maps contains all possible kneading invariants for a given modality and signature (sequence of signs of the slopes) and the isentropes (loci of constant topological entropy) are proven to be connected. This was used to prove that more generally, the entropy is monotone (more precisely the isentropes are connected) for full families of polynomial maps, a recent result of Bruin and van Strien following:

- the special case of bimodal maps treated by Milnor and Tresser (proving a conjecture of Milnor that had resisted for some time),

- and the most important case of unimodal maps treated first by Douady-Hubbard, Milnor-Thurston, and Sullivan, who hereby filled the hole in a false proof in the paper of Metropolis, Stein, and Stein where these authors used what was later to be called kneading sequences to describe what they believed to be a universal topological property of "reasonable" families of unimodal maps.

Monotony means in fact that the partial order on the complexity of dynamics provided by the use of kneading theory corresponds to a similar partial order for full polynomial families (for instance, the complexity is non-decreasing for the quadratic family): probably the first step in getting a global view of the dynamics in these families of maps.

On specificities of kneading theory

Preliminary warning. In what follows we have stayed as far as possible from the topics and methods used in the () article (that is excellent but, if anything and some critics must be made, has rather too much proofs and examples on the aspects that are covered and is arguably badly targeted). To give more chances for these notes to be useful, we have indicated some possible directions of research but so many authors have had abundant contributions, such as Hao Bai Lin and many others, that a careful inspection of the literature would be required to be sure that the questions raised here are still open. One type of question that is probably quite hard to answer, and that has unknown expected benefits in term on understanding dynamics, is to provide an example of a kneading sequence such that the orbit of the unique critical point is dense in the interval between the second and the first image, a property known to be generic (Brucks, Diamond, OteroEspinar, Tresser) and of full measure (Brucks and Misiurewicz) for tent maps.

Associating symbol sequences to arbitrary maps

Given some space \(S\), an idea (close to what is called coarse graining in some circles of physicists, and some may even say that this is nothing but coarse graining), consists in splitting \(S\) into subsets \(S_i\), with \(i\in \mathcal{I}\)where the index set \(\mathcal{I}\)is finite or not, and associate to any point \(p\in S\) its address \(A(p)=i\, \,\leftrightarrow \,\, p\in S_i\). Assume then that we are interested in some map \(f:S\to S\), invertible or not: then one can associate to any \(p\in S\) an itinerary that is the sequence of addresses \(A(f^n(p))\) where:

- \(n\) runs in Z if \(f\) is invertible,

- \(n\) runs in Z\(^+\) if \(f\) is not invertible (or the past somehow does not matter).

Other means exist to associate symbols to orbits for flows or more generally to the action of groups or semi-groups that are either discrete or continuous, but let us only consider here the addresses that can be constructed for maps. While specialists in abstract symbolic dynamics consider primarily this subject as the action of the shift map (invertible or not) on some invariant closed set, for specialists of maps (or specialists of flows who use maps as tools to study flows), symbolic dynamics is a mean to learn more about their primary object of study and they proceed as we have indicated (although flows sometimes permit types of symbolic dynamics of a different nature as in the original example devised in 1898 by Jacques Hadamard to study the geodesic flow on surfaces of negative curvature: to understand the progress that has been done in a bit more than 100 years, recall that the thesis of Marston Morse in 1921 mostly consisted ion using symbolic dynamics to prove the existence of recurrent non-periodic orbits for such geodesic flows). For us here, symbolic dynamics means what one could call applied symbolic dynamics i.e., the application of abstract symbolic dynamics as we have indicated. Probably the most prototypical example of applied symbolic dynamics concerns maps that are structurally stable, so that a small perturbation of the map is topologically conjugate to the flow\[f'=h^{-1}\circ f\circ h\] for some homeomorphism if \(f'\) is close enough to \(f\) in the \(C^1\) topology. Then the image of \(S\) under \(h\) provides the same symbolic for \(f'\) that \(S\) provides for \(f\). Families of maps are then out of the picture since deformations do not change anything.

Why bother with kneading theory and interval maps?

The situation is quite different for natural parametrized families of interval maps. Hence it seems a priori well founded to have an article on the specific symbolic dynamics for such objects, called kneading theory just as the important subset covered by the seminal paper of Milnor and Thurston on his topic. But do parametrized families of interval maps deserve a lot of special interest that would in turn cascade to kneading theory? We will only cover here one story that has been of interest both in pure mathematics and in applivcation to many other fields. Such maps have proved to be useful models in particular because of Universality properties, in particular at the transition from zero to positive entropy. It so happens that the dynamics of flows that cannot find analogs in flows in dimension three is quite complicated, with the complexity growing very fast with dimension since tools like those used in dimension three are missing (for instance the braid type of an orbit cannot anymore in dimension above three) and the richness of possible behavior seems to explode if the dimension grows in a fundamental way, i.e., so that some phenomena happen that cannot be modeled by equations in lower dimension. It so happens also that if one can expect a volume contracting flow to be a reasonable model for a system that goes from regular to complicated when varying a parameter, then a Poincaré map is often close to a one dimensional map as long as one has not yet gotten into the chaotic regime but up to the transition to complicated dynamics. Then in a great number of cases, the transition to complicated dynamics, say to positive topological entropy, is not only like in smooth interval maps by a cascade of period doubling, but also generically some deeper universality holds true. As conjectured by Coullet and Tresser in 1977, the sequence of ratios of lengths of consecutive parameter windows where the periods \(2^n\) and \(2^{n+1}\) are stable converge toward about 4.669 for many natural systems and flow models for the such systems. This number (in fact 4.669201609101990... ) was first observed and conjectured to be universal in generic families of unimodal smooth enough interval maps(i.e., conjectured to hold true independently of the family of maps as long as said family is generic, and more precisely has a quadratic extremum), by Feigenbaum and by Coullet and Tresser. Furthermore, this universality phenomenon (that resembles second order phase transition in Statistical Physics) was conjectured by both groups to be explained (like in Statistical Physics, the so called Kadonoff-Wilson theory) by the convergence of a renormalization group acting on interval maps (or more general maps) and such that for maps at the transition to positive topological entropy, the renormalization has a single attractor for generic maps. So to speak, what is for now best known about the transition to chaos in natural systems is often explained by first studying interval maps and then explaining that extra dimensions do not spoil the picture. Of course, only a small proportion of all natural systems go to chaos the way interval maps do, but one has many examples in many branches both of engineering and of many natural sciences where the transition is by a cascade of by period doubling bifurcation like in families of interval maps and the ratio has been measured to be about 4.669. It is noticeable that the one dimensional story is now covered by rigorous mathematics, the first part of it can be expressed using the language of kneading theory, and that progress toward a rigorous treatment is currently being made by de Carvalho, Lyubich, and Martens for diffeomorphisms of the plane following some pioneering work made on more topological related issue by Holmes and Wiliams and by Hall who has unfortunately modestly omitted his work that started with his Ph. D.

Associating symbol sequences to members of families of interval maps

Unimodal maps

We start with maps on \(I=[0,1]\) such that 0 and 1 are mapped to 0 that is thus always a fixed point to simplify the discussion. We consider in fact one parameter families \(f_\mu\) of maps that are smooth, depend smoothly on \(\mu\) and have a single critical point \(c\in (0,1)\) that we will assume to be fixed, again to avoid superfluous generality. The originality of kneading theory resides then in that, independently of the value of the parameter \(\mu\), \(S\) consists in three pieces, and more precisely the two semi-open intervals \(L=[0,c)\) and \(R=(c,1]\) corresponding respectively to the addresses \(L\) and \(R\) and a third piece reduced to the single point \(c\) that corresponds to the address \(C\). Metropolis, Stein, and Stein considered in the early 1970's the kneading sequence \(K(f_\mu)\) defined as the itinerary of \(f_\mu\) that runs forever if \(c\) is not periodic and stops at the first \(C\); i.e.,, the kneading sequence is the truncated itinerary of the critical value, where truncation only happens if one reaches a \(C\). These three authors also noticed that for many smooth families, \(K(f_\mu)\) seems to evolve with \(\mu\) in a way that does not depend on the particular choice of the family, a sort of symbolic universality that somehow correspond to the general order structure on the interval, as does the topological universality expressed by the Theorem of Sharkovsii and generalizations of it that describe the forcing partial order between permutations induced by periodic orbits. There is a natural order on itineraries that can be defined as follows: one associates to an itinerary \(\textbf{I} (x)\) (or to a truncated itinerary that stops at the first \(C\) if any), a number \(\mathcal{V}\)\( (\textbf{I} (x))\) written in base 3 that starts \(.0\) and whose successive symbols are obtained by reading \(\textbf{I} (x)\) and:

-Writing a 2 if one reads a \(R\) and the previous symbol in \(\mathcal{V}\)\((\textbf{I} (x))\)(perhaps before the dot) was a 0 or if one reads an \(L\) and the previous symbol in \(\mathcal{V} (\text{I} (x))\)(perhaps before the dot) was a 2,

-Writing a 0 if one reads a \(R\) and the previous symbol in \(\mathcal{V}\)\((\textbf{I} (x))\)(perhaps before the dot) was a 2 or if one reads an \(L\) and the previous symbol in \(\mathcal{V} (\text{I} (x))\)(perhaps before the dot) was a 0,

- Writing a 1 if one reads a \(C\).

Consider now the unimodal \(Stunted Sawtooth Map\) with signature (succession of signs of slopes where defined) equal to \(+-\), \(H_\mu\) that is like one of the \(f_\mu\)'s mentioned before except for smoothness that will hold only almost everywhere for the family \(H_\mu\). First \(H_1\) is defined by:

<math eqn:FullSSM>

\left\{ \begin{array}{lcl} & & 3x\qquad\quad\,\,\, \textrm{if} \,\,x\in [0,\frac{1}{3}]\\ H_1(x) &=& 1\qquad \,\,\,\,\,\,\quad \textrm{if}\,\, x\in [\frac{1}{3},\frac{2}{3}]\\ & & 3(1-x)\quad \textrm{if} \,\,x\in [\frac{2}{3}, 1]\end{array} \right. , </math> so that one next defines more generally \(H_\mu(x)= \min (\mu, H_1(x))\). One can then check that any admissible itinerary, i.e., any sequence \(\mathcal{I}\)that could be an itinerary \(\mathcal{I} (x)\) of some point under any map such as \(f_\mu\) or \(H_\mu\) is indeed the itinerary of some point \(x_{\mathcal{I}}\) under \(H_1\). In fact all truncated itineraries are admissible itineraries as a consequence of simple monotony considerations. As another bonus of the other structure, one can easily check that a truncated itinerary \(\mathcal{I}\) is a kneading sequence of some \(H_\mu\) if and only if \(\mathcal{V}\)\((\textbf{I})\) is bigger than the value of any shift of that sequence. At last, still as a bonanza of the order properties on the interval, one can check that \(\mathcal{V}(K(H_{\mu_1}))\geq\mathcal{V}( K(H_{\mu_2}))\) if and only if \(\mu_1\geq K(H_{\mu_2})\). Metropolis, Stein, and Stein thought that they had proven that \(\mathcal{V}(K(f_{\mu_1}))\geq\mathcal{V}( K(f_{\mu_2}))\) if and only if \(\mu_1\geq K(H_{\mu_2})\) when \(f_\mu\) is the quatdratic family \(f_\mu(x)=4\mu x(1-x)\), but that proof was false as all attempts so far to prove this result with help whatsoever from complex analysis nor from complex dynamics, but many proofs have been provided without such technical constraints. They noticed however that there was a essential qualitative universality in the way such families evolve from regular dynamics to more complex dynamics and in particular studied the formula corresponding to period doubling. This and more generally renormalization in general was studied, in a language poorly understood by most others by the group of Christian Mira that was more aware than many of the work of Myrberg and others and early on understood the power of computers as an help to study dynamics. In more easily communicable way, Derrida, Gervois, and Pomeau developped what they called a star product, that corresponds in fact to the embedded box structure of Mira et al. and in more modern terms to renormalization seen at the p[urely symbolic level. The same operation with the same star symbol was later independently used by Jonker and Rand who studied the structure of the \(\Omega\) set as reported in the article. The easy monotony of the family \(H_\mu\) was used to establish the monotony in other piecewise affine families by Brucks, Misiurewicz, and Tresser. The ease produced by stunting was first used to study endomorphisms of the circle by Chenciner, Gambaudo, and Tresser and kneading was first studied using this ease by Mumbrú in his thesis where he corrected some errors of MacKay and Tresser in their study of the bifurcation structure of the cubic families.

Multimodal maps

The order structure reported above can be restated in a weaker form as the monotony of the topological entropy as \(\mu\) varies (although of course much more gets proved, and in fact the monotony of the kneading sequences). While kneading theory was invented and used for maps with finitely intervals of monotony or multimodal maps by Milnor and Thurston in their fundamental paper as finely reported in the article, the general Stunted Sawtooth Maps approach was introduced by Milnor and Tresser in order to generalize the monotony properties to multimodal case. Early on, Milnor had proposed to generalize the monotony of entropy in the quadratic family to the property that isentropes (parameter sets with a given entropy for the map with parameters in that set). In the Milnor and Tresser paper, the connectivity of the Stunted Sawtooth Maps family with a given signature (with \(n+1\) signs for modality \(n\)) was proved for any modality. This result was used by these authors to prove a similar property for cubic maps, a result that was recently generalized to all full polynomial families (also on the basis of what happens for the Stunted Sawtooth Family) by Bruin and van Strien. In the multimodal case, a generalization of the star product was studied by Brucks, Galeeva, Mumbrú, Rockmore, and Tresser: in the general case, the main issue is to understand how permutations can be written as products of permutations realizable by unimodal maps wit prescribed signatures.

Beyond multimodal maps

Continuous interval maps that are not multimodal

Not much if any has been done for maps that fail to be multimodal because they present infinitely many segments of monotony: these appear as reasonable subjects of study since most infinitely renormalizable dynamics that can be realized by maps with entropy zero can only be realized by such non-multimodal maps that were investigated by Glendinning, Los , Otero-Espinar and Tresser and by Tresser and Wilkinson in the smooth case. A vast domain of investigation thus remains quite unchartered.

Discontinuous interval maps (a): circle maps and rotation compatible renormalization

Starting where we have left continuous maps, there are important discontinuous maps with infinitely many discontinuities such as the Gauss map \(G\) that is associated to continued fraction expansion (i.e., for any real \(x\in [0,1]\) with continued fraction expansion \([a_1, a_2, a_3,\dots]\) the image \(G(x)\) of \(x\) under the Gauss map is the real number in \([0,1]\) with continued fraction \([a_2, a_3,a_4, \dots]\) . For maps with infinitely many discontinuities, kneading methods (symbolic dynamics corresponding to the interval of monotony) could be used but this again is mostly unexplored except perhaps for a few special maps. Similarly, the contribution of kneading theory to interval exchange transformations seems to be marginal, except in the case of rotations (interval exchange transformations with two intervals; indeed the main generalization of rotation when it comes to symbolic dynamics is by considering rotations in higher dimension rather than exchanges of many pieces): the possible itineraries are Sturmian Sequences that were introduced in abstract symbolic dynamics by Morse and Hedlund. These are sequences such that any two blocks of length \(n\) either have the same number of \(0\)'s or have number of \(0\)'s that differ by 1. Since interval exchanges are invertible, one can use infinite or bi-infinite sequences with itineraries that go both ways (listing iterates under the map and its inverse). Beside itineraries of rotation, Sturmian Sequences present exceptional cases such as the sequence \(0^\infty 10^\infty\): it has been shown that the set of Sturmian Sequences is exactly the set of possible itineraries for homeomorphisms of the interval (Siegel, Tresse, Zettler). Back to rotations, one can also code with any two intervals that cover the circle with no overlap and characterize the sequences that can be obtained that way, a problem that find its origins in studies of the heart (Siegel, Tresse, Zettler and Alsedà, Gambaudo, Mumbrú). Rotations play for circle homeorphisms about the same role that maps with uniform absolute value of the slope play for endomorphisms of the interval: they are models up to semi-conjugacy when there are no periodic orbits. Homeomorphims of the circle are important object of study because of considerations that largely exceeds issues related to the transition to positive topological entropy: they are the basic models foe oscillations process of all sorts and permit for instance to study the phenomenon of frequency locking that was first observed and experimented upon by Huyghens in the seventieth century and that explains among other things the stability of many instance periodic phenomena and from the abstract point of view why normal forms sometimes fail to capture all the dynamics.

Renormalization of a type different from what corresponds to a star product can be formulated for circle homeomorphisms, with application to a variety of important problems: for instance one considers the arc between a point and its image (one of the two complementary arcs in the splitting that leads to Sturmian Sequences) and then consider the first return map to that arc and renormalize to a full circle preserving or not orientation. Depending on the choice that one makes of the interval on which one considers the first return map, one gets a map on the rotation number (the asymptotic proportion of 1's in the itineraries that are possible for the map) that in any case decreases the denominator in the case when the rotation number is rational. The most studied case is when the map on the rotation number is exactly the Gauss Map. In any case, we have what one can call a rotation compatible renormalization, an operator that preserves the set of rotations if one consider that when one has a fixed point for the map, the renormalization has reached a fixed point. The rotations are indeed attractors for rotation compatible renormalization acting on smooth enough circles diffeomorphisms, a part of Herman's theory discovered by Deligne, and the rate of convergence is furthermore fast enough to make the diffeomorphism conjugate to a rotation when the rotation number satisfies a Diophantine condition (Herman, Deligne, Yoccoz, Katznelson-Ornstein, Khanin-Sinai). Period doubling renormalization, and more generally the renormalization associated to a star operation can also be considered as the operator that generates a rescaled first return map: while there are two return times for rotation compatible renormalization, there is only one return time for the renormalizations that correspond to a star product (more complicated renormalization have since been introduced for endomorphisms of the interval by Lyubich, Milnor, van Strien, Nowicki, and many others). Using Lorenz like maps, Gambaudo, Procaccia, Thomae, and Tresser have pointed out the return maps are a common mean to understand renormalization of the star and rotation type.

Discontinuous interval maps (b): Lorenz maps and the associate boundary of zero topological entropy.

The class of discontinuous interval maps that have been studied using symbolic dynamics go much beyond invertible maps, and the case that has been most studied is probably the case of generalized Lorenz maps, that contains maps that either have singularities, zeroes, or some slopes in between as limits on both sides of the discontinuity. These maps arise in the study of flows in three dimensions that are close to have two homoclinic connections on a saddle point, and indeed the class of flows can be enlarged so that the derivative can have any sign away from the singularity, the classical Lorenz case being when all slopes are positive. It is only in the Lorenz-like case with positive slopes, say the \(\{+,+\}\) case, that kneading theory has been greatly used so far although all cases seem to be of interest, revealing many types of transitions from zero to positive entropy (Glendinning, Los, Tresser). In the \(\{+,+\}\) case, the study is somewhat simplified by the fact that the values is more directly associated to itineraries because the slopes are positive. In the case of maps with exactly one slope on each side of the discontinuity, it is known that the topological entropy is positive if and only if some interval is made of points that have two pre-images. In the case when the slope is zero on both sides of the discontinuity, all maps with zero entropy that have more than Sturmian Sequences as itineraries can be renormalized using rotation compatible renormalization or period doubling renormalization. After the renormalization, one gets a map that only has Sturmian Sequences as itineraries or that can be renormalized using rotation compatible renormalization or period doubling renormalization. The process stops at a map that only has Sturmian Sequences in its set of possible itineraries or continues forever. No fully symbolic proof of the characterization of Lorenz like maps with zero entropy has been provided. Similarly, computation of entropy that would use only symbolic manipulation have never been provided: it seems that the interplay between symbolic dynamics and the maps (endomorphisms of the interval or the circle, Lorenz map, etc.) is much easier to use.

Degree \(d\) maps of the circle, kneading, and conjugacy.

We finish now this complementary overview by discussing briefly degree \(d\) maps of the circle with \(|d|>1\) and then reconsidering again the whole story. The degree \(d\) maps for \(d\not = 0\) are also special cases of discontinuous interval maps, and one can say that in this perspective (and only in this perspective) the Lorenz-like maps with slope everywhere positive interpolate between degrees one and two (a way of talking that infuriates many topologists). Anyway,assuming that some iterate of the map has slope everywhere greater than one in absolute value, the splitting into \(|d|\) maximal monotone branches gives a natural alphabet that, with the slope property that we have imposed, has the nice property that no pair of distinct points can have the same itinerary. A further small exercise consists in proving that all sequences written with the \(|d|\) symbols is realized. Consider then two maps \(f\) and \(g\) with the same degree and slope everywhere higher than 1(or both smaller than -1) for high iterates and let \(h(f,g)\equiv h:T^1\to T^1\) be the map from the circle (seen as the one-dimensional torus) to itself defined from \(f\) and \(g\) as \(h(x)= \mathcal{I}\)\(^{-1}_g\)\((\mathcal{I}\)\(_f\)\((x))\,\) where the inverse of the itinerary map is well defined since every itinerary is realized and is furthermore realized by a single point. A last effort will lead us to prove that \(h(f,g)\) is continuous, which tells us that that any two maps with degree \(d\) that have an iterate with slope everywhere greater than one in absolute value are topologically conjugate, a result not that sample in the late nineteen sixties (Shub).

Coming back now to multimodal maps, we are in a case where all turning points and the two extreme points have itineraries whose full collection essentially determines the set of possible itineraries. In the best cases, this allows one to construct conjugacies between maps, but we know by a theorem of Milnor and Thurston that is recalled in the article that two such maps are semi-conjugate to some well defined map that has a constant value of its slope if topological entropy is positive. In some sense the extreme sets of itineraries implied by the full set of kneading sequences is somewhere between what one finds in the Stunted Sawtooth Maps for the maximal set and in the map (perhaps with different ititeraries as star products are to be cleaned of) that has constant absolute value of the slope.

Conclusion

Contemplating again the whole story as told in the article by Toby Hall and in these complementary notes, let us point out that is not so original to have a great deal of information contained in the symbolic dynamics associated to maps: for interval maps and related maps, one can go much further and in particular use the same addresses set to code all the members of parameterized families. In particular, kneading theory permits to discuss conjugacy and semi-conjugacy issues and furthermore provides handy tools to discuss aspects of the monotony of behavior with parameters and in particular the topological aspect of universality of the transition from zero to positive topological entropy.

This review is written by Charles Tresser after quickly consulting over the phone with a few colleagues on what he planed to cover. Assuming that what is most important is what is what one remembers, no notes nor paper was consulted except to check that there is no easily quotable theorem on star product for multi-modal maps had. The author of the review apologizes for the strong bias toward citing his own work that resulted from this choice of preparation for reviewing; anyway, most of the most basic literature was thoroughly covered in the article of Dr. Toby Hall.