The role of genericity in the history of dynamical systems theory
Abstract and Keywords
This article examines the role of genericity in the development of dynamical systems theory. In his memoir ‘Sur les courbes définies par une équation différentielle’, published in four parts between 1881 and 1886, Henri Poincaré studied the behavior of curves that are solutions for certain types of differential equations. He successfully classified them by focusing on singular points, described the trajectories’ behavior in important particular cases and provided new methods that proved to be extremely useful. This article begins with a discussion of singularity theory and its influence on the first definitions of genericity, along with the application of the notions of structural stability and genericity to understand dynamical systems. It also analyzes the Smale conjecture and how it was proven wrong and concludes with an overview of changes in the definitions of genericity meant to describe the ‘dark realm of dynamics’.
In mathematics, general descriptions of mathematical beings in a given universe often engage a tension between the generality of the description and the relevance of the properties chosen to be described. In the case under study, mathematicians’ aim, in their attempt to classify or, better, to provide a general description of dynamical systems, was to find a compromise between the two poles: generality and relevance. The notion of genericity, along with its different mathematical definitions, plays a fundamental role in the history of dynamical systems theory.
In his memoir “Sur les courbes définies par une équation différentielle,” published in four parts, from 1881 to 1886, and known as the starting point of the study of dynamical systems, Poincaré analyzed the behavior of curves that are solutions for certain types of differential equations.1 He succeeded in classifying them by focusing on singular points, described the trajectories’ behavior in important particular cases and provided new methods that proved to be extremely useful. Some historians have already analyzed these works: Christian Gilain in the context of research on differential equations in Poincaré’s time,2 Anne Robadey from the perspective of generality in a different sense from the one exposed here,3 and myself, in relation with stability issues.4
Despite its fundamental importance, this study was far from describing the classes of differential equations themselves. Over 20 years after this first breakthrough, during (p. 300) the 1908 International Congress of Mathematicians in Rome, Poincaré announced his prospects for future research on differential equations as follows:
Much has already been done for linear differential equations, and one only needs to perfect what was started. However, concerning nonlinear differential equations, developments have been too modest. The hope of integrating with known functions was forsaken long ago; we therefore need to study in and of themselves the functions defined by differential equations, starting with an attempt to systematically classify them. A study of the growth mode in the vicinity of singular points will most probably provide the first elements for such a classification, but we will not be satisfied until some group of transformations has been found (such as the Cremona transformations), playing—vis-à-vis the differential equations—the same role as that played by the group of bi-rational transformations in the case of the algebraic curves. Then, we would be able to place in a single class all the equations derived from one of them by transformations. We would have the analogy with an existing theory to guide us: that of bi-rational transformations and the genus of an algebraic curve.5
The above quotation clearly demonstrates that Poincaré’s wish was that this classification could have the same general character as that of the algebraic curves classified according to their genus. We need not go into the details of the analogy that Poincaré points to in this quote, but we ought to underline three elements that will play a part in this chapter.
First, Poincaré points to a shift in terms of objects and investigative means. In elementary cases, one is accustomed to solving a differential equation by exhibiting a formula built from elementary functions (such as exponentiation or arctangent). Usually, a single formula describes a family of solutions, since it contains arbitrary constants. However, in all but the most elementary cases, this family fails to encompass all solutions: “singular” solutions are, more often than not, left out. In more complex situations, the functions that are solutions of a given differential equation usually cannot be expressed by simple formulas, hence the very meaning of what it means to solve a differential equation has to be altered. In his early work on the qualitative study of differential equations (1881–6), Poincaré showed that a new viewpoint allowed for relevant information to be gained even when explicit formulas are beyond reach. This new and qualitative viewpoint relies on two significant shifts. First, Poincaré chose to focus on the curves that are defined by the solutions, hence a shift from the analytic to the geometric. Second, the theory does not deal with individual solutions (seen as functions or as curves), but with the system of all solutions. In particular, singular solutions must be taken into account from the beginning since they provide the first elementary pieces of information on the basis of which the set of solutions can be investigated as a geometric object.
Second, when it comes to studying the sets of solutions to all differential equations of a given type, some classification principle has to be devised. One needs to find a way of capturing technically the idea that two different sets of solutions are equivalent from the qualitative viewpoint. If that could be achieved, all equivalent sets of solutions could (p. 301) be “sorted into a single class,” leaving two tasks for the mathematician: identifying the relevant geometric properties that tell to which class a given differential equation belongs; and enumerating the different classes thus obtained. To serve this purpose, Poincaré suggests that groups should be brought into the picture: two differential equations (or two sets of solution curves) would be considered as equivalent if and only if they could be transformed one into the other by a transformation from a given group. In this framework, finding the right equivalence relation boils down to finding the right transformation group. More often than not, there are several possible groups to choose from, and hence several (possible) equivalence relations: the larger the group, the coarser the equivalence relation and the cruder the classification.6 This is one of the aspects for which the analogy with the theory of algebraic curves is enlightening. A fine-grained classification of algebraic curves relies on the group of algebraic isomorphisms. Using the larger group of birational transformations entails a loss of information (curves which are algebraically inequivalent can be birationally equivalent), but this is the price to be paid for a convenient classification: algebraic curves (more precisely smooth, complex, projective algebraic curves) are birationally equivalent if and only if they have equal genus; the genus being a numerical invariant which can easily be computed, the coarser classification proves tractable. This coarse classification can be seen as a first step on the way to the finer classification.
Third, in this quotation, Poincaré presents a rather loose analogy between the theory of algebraic curves and that of systems of curves defined by differential equations. He merely suggests that a very general strategy, which had been successfully used in the case of algebraic curves, should be used in another theory. In this chapter, we will see several cases in which proof strategies and concepts were introduced in a given theoretical context by mathematicians who explicitly relied on an analogy with another theoretical context. This phenomenon is fundamental for the understanding of the relationship between the various concepts of genericity that we will encounter.
In this study, we will focus on the second half of the twentieth century, hence on a much later phase of the development of the qualitative study of differential equations. The theory would be re-christened the “theory of dynamical systems.” Sets of solutions would be tackled in the language of function sets, or function spaces. The progressive discovery of the complexity of systems of solutions would lead to several classification attempts, based on different groups; on different notions of equivalence or “sameness” or “proximity;” (p. 302) on different “suggesting sciences” (differential topology, then measure theory). All these elements fit the general scheme outlined by Poincaré in the above quotation.
However, one fundamental element is not related to this description. As we shall see, mathematicians would soon set themselves the more tractable task of classifying “almost all” solutions. To implement this strategy, two concepts must be defined: that which characterizes the subset of solutions whose description is tractable; and that which gives a technical meaning to “almost all.” Our historical narrative will document the use of several pairs of concepts. From a more epistemological viewpoint, the core issue is that of the interdependence of the two concepts, and of the compromises that must be made: the subset picked out by the first concept must be small enough to exclude cases which are beyond our descriptive capacities, but large enough to be considered “generic,” that is, to satisfy the “almost all” requirement. This dialectic can be thought of as a productive tension in the development of the theory of dynamical systems; a tension which the historical development of the concept of genericity helps one to understand.
10.2 The influence of singularity theory on the first definitions of genericity
At first, the qualitative study of differential equations Poincaré proposed was explored by few mathematicians,7 the best known examples being Hadamard and Birkhoff. Dynamical systems, as mathematical objects, were defined in 1927 by Birkhoff,8 and his definition already implied a qualitative view.9 Nonetheless, it was only in the fifties that the theory gained a new impulse, mainly with the classification efforts which will be analyzed throughout this chapter.
The establishment of dynamical systems as a theory is related to classification goals, in close relationship with what was to be later called singularity theory of differentiable mappings. One of the major aims of the theory was to classify mappings by the type of their singularities. The classification of algebraic curves, mentioned by Poincaré, succeeds in explaining the type of all such curves by their genera. In singularity theory, mathematicians also tried to classify all objects of a given universe, for instance mappings from Rn to R. However, they only obtained a classification of almost all objects of this universe. In dynamical systems theory, as the curves have much more complex behaviors, the maximum one could expect to obtain is a classification of almost all objects of a given universe. So, prior to classification, we must know if the classifiable objects are almost all objects in a universe.
(p. 303) The aim of that which would later be called singularity theory was to obtain a picture of the kind of singularities that an arbitrary function from Rn to Rp can have. Since it is impossible to characterize all functions in this way, the strategy used was to determine the kind of singularities a certain prototype function can have for each universe (obtained for fixed values of n and p), and then show that any function in such a universe can be approximated by the prototype. This means that we classify the prototypes and show that any being in the universe can be approximated by one prototype. It was René Thom who expressed this last property of prototypes in terms of genericity. In the search for generic properties, the question is to establish that almost all (and not necessarily all) beings are classifiable.
Beginning in 1925,10 Marston Morse studied the singularities of functions from a manifold M ⊂ Rn to R. He found that any function of this type can be approximated by a function having only isolated singular points.11 A smooth real-valued function on a manifold M was later called a “Morse function” if it had no degenerate critical points12. Nowadays, the most important result of Morse theory says that almost all functions (in C2 topology) are Morse functions. Besides, as Morse functions have only one type of singularity, which may be easily characterized, it is possible to obtain a satisfactory description of almost all functions in the universe of real-valued functions.
In a paper published in 1955, on the singularities of mappings from R2 to R2, Hassler Whitney defined an “excellent” mapping as being a mapping for which all singular points are “folds” (analogous to singularities of Morse functions) and “cusps,” which are well characterized types of singularity.13 Then, he showed that, arbitrarily close to any mapping of the plane into the plane, there is an excellent mapping. This means that any mapping of that type can be approximated by a mapping having only well-characterized types of singularities (see Fig. 10.1).
The following year, in an article presented in the Bourbaki Seminar,14 Thom proposed studying, with the same methods as Morse and Whitney, the mappings from Rn to Rp and characterizing them by the type of their singularities. It was thus necessary to analyze mappings that can have well-determined singularities and that can approximate an arbitrary function in the space of functions from Rn to Rp. The question is to find classifiable prototypes and to show that any function can be approximated by one of them. So, the properties of the prototypes would be called generic. But we still do not know exactly what this means.
Thus, Thom defines a generic property P of a mapping of class Cm from Rn to Rp as a property verifiable by all functions belonging to the space of functions of this type, except for a “thin” subset of that space. This fact can be viewed as an extension to differentiable (p. 304)
structures of the notion of genericity used in algebraic geometry. In this theory, a generic property was already defined as a property that can be satisfied for all points of a space, except for the points of a thin sub-manifold of that particular space. In the spring of 1952, while spending a year in Princeton University, Thom had, in his own words, “a striking conversation” with Claude Chevalley who suggested introducing the notion of genericity used by the Italian geometers to the world of differential structures.15
In a subsequent article, published a few months after the Bourbaki one, Thom explicitly notes the parallelism between the definition of genericity presented in his first article and the one employed in algebraic geometry, with the difference that, in the context of differentiable mappings, the functions studied constitute a functional space that is a Baire space and the exceptional thin sub-manifold is replaced by a closed subspace without interior points.16 But Thom was not completely satisfied with this terminology. Despite this fact, the adjective “generic” has been disseminated with the meaning that generic transformations are those that can approximate any transformation. This notion of “approximation” had a precise definition. A generic property was then known as a property that is satisfied by the elements that constitute an open and dense subspace of the domain, which is the complement of a closed subspace without interior points.
A successful classification program in the theory of singularities of differentiable functions should be able to define classes of functions with the following characteristics: (1) each class is sufficiently particular to be geometrically well described by the type of its singularities; and (2) such classifiable functions are sufficiently general to include “almost all” functions in the sense that they constitute an open and dense subspace of the domain of all functions.17
(p. 305) For differentiable functions, these two conditions express the compromise mathematicians were searching for, between the relevance of the properties under study (given by the type of singularities) and the generality of these properties or its genericity (the size of the subset of entities satisfying these properties). The type of the singularities of a function successfully captures its qualitative aspect and, if the two conditions cited above are fulfilled, we can conclude that almost all functions are classifiable by their singularities. The importance of this conclusion justifies the search for classifiable functions that form a generic subset in the domain of all functions. A similar goal would motivate the development of dynamical systems theory in the fifties and the sixties.
10.3 Globally understanding dynamical systems through the notions of structural stability and genericity
During its renewal at the end of the fifties, the style that the theory of dynamical systems acquired was greatly influenced by the program of classifying singularities of real functions, and vice-versa. In his doctoral thesis, David Aubin shows how the practices of what he calls “applied topologists”— working at IHES (Institut des Hautes Études Scientifiques) under Thom’s leadership and having inherited from the Bourbakists the interest in structures—influenced the style acquired by qualitative dynamics, notably in the pioneering works of Peixoto and Smale.18 An understanding of the way this style was transmitted involves, as we will see later, the direct collaboration among the three mathematicians, not only at the IHES but also at the Brazilian IMPA (Instituto de Matemática Pura e Aplicada).
The Brazilian mathematician Mauricio Peixoto was convinced that the main goal of the mathematics of his times was to classify mathematical objects, with emphasis on their structures, and by means of equivalence relations between them.19 He thought it would be a useful challenge to express the theory of differential equations in a set-theoretic language. From his point of view, the suggestion given in Poincaré’s quotation (transcribed in the introduction) had to be fulfilled with notions extracted from set theory.
Poincaré’s and Birkhoff’s work was certainly the point of departure for such a study, but in order to express their theory in a set theoretical basis it was still necessary to introduce two new elements:20
A) A space of differential equations, or dynamical systems, possessing a topological structure.
B) A notion of qualitative equivalence between two differential equations (analogous to Cremona transformations as was claimed by Poincaré).
Both requirements were fulfilled, primarily in two articles: the first written in 1958 and published in 1959 and the second published in 1962.21 Peixoto defines the space of dynamical systems by considering a dynamical system as a point of a Banach space, (p. 306) and proposes that an equivalence relation between two systems in this space should be a homeomorphism, transforming trajectories of one system into trajectories of the other. This last definition is inspired by the work of Andronov and Pontryagin.
In 1937, these two Soviet mathematicians had published a paper called “Systèmes grossiers,”22 in which they studied dynamical systems defined in a two-dimensional space and proposed that the trajectories of two systems should be considered equivalent if they could be transformed into one another by means of a transformation that is a homeomorphism (with the additional condition that this transformation is close to the identity transformation23). A system is called “grossier” (which means “coarse” and can also be translated as “robust”) if its trajectories remain qualitatively similar after a perturbation in the definition, and the homeomorphism is precisely the transformation considered in order to maintain trajectories that are “qualitatively similar.”
The importance of the coarseness property consists in the role it plays in modeling physical systems. If a system is not coarse, or robust, its fundamental properties are easily lost after a small perturbation. As mathematical models are just idealizations of physical realities, we cannot avoid perturbations in the definition of the mathematical system. Thus, a coarse system is a good candidate to serve as a model for a physical situation.
Around 1950, as Dahan-Dalmedico showed,24 “grossier” was renamed “structurally stable,” following a suggestion of Lefschetz. By this time, there were some researchers working on the subject in Princeton, and that led Peixoto to join them in 1957.
In 1952, a mathematician of Lefschetz’s team, De Baggis, managed to provide the demonstrations that were lacking in Andronov and Pontryagin’s article and alleviated some of the requirements they had recognized as necessary. He opens his article by claiming that:
In the study of nonlinear problems it is difficult for the mathematician to find rich classifications of nonlinear systems which are sufficiently homogeneous in their properties to yield an interesting theory.25
He explicitly refers, in this quotation, to the productive tension we claim is present in the efforts undertaken, during the fifties and sixties, by mathematicians trying to obtain some kind of classification of dynamical systems.
The difficulty of this research direction is due to the enormous variety of behaviors of trajectories of dynamical systems in the nonlinear case. In order to provide general descriptions it is necessary to impose some restrictions that make it possible to choose a subset of interesting systems in the universe of nonlinear systems. But what are the interesting properties that can be used to restrict this enormous universe?
(p. 307) Issues of physical relevance can provide a clue to the right concept of structural stability. From the standpoint of applied mathematics, a supplementary advantage of such a concept is that it would solve the problem of legitimacy of modeling raised by the Soviet mathematicians because, even if we know a model is never exact, it can always be approximated by a structurally stable one, which means it can be approximated by another model, similar to the original, that does not lose its main properties after perturbation.
Andronov and Pontryagin attempted to give a mathematical description of two-dimensional structurally stable systems, proposing that the essential features of these dynamical systems that are able to be preserved under small perturbations are: (1) it has a finite number of singularities, all of which are simple; and (2) no trajectory of the system goes from a saddle point (that is a type of singularity) to another saddle point. These features express the topological character of the set of trajectories of a dynamical system defined on a two dimensional manifold. In his article of 1959,26 Peixoto showed that structurally stable systems, having Andronov and Pontryagin’s features, form an open and dense subset in the space of all systems defined on a sphere. It is the first general result in the theory of dynamical systems.27 Even though it only holds for two-dimensional systems defined on a specific surface, Peixoto’s theorem is a general result, since it succeeds in describing the relevant features of almost all two-dimensional dynamical systems.
In the following years, the result was reformulated in the mode of singularity theory. Smale was introduced to Peixoto in the autumn of 1958, during his postdoctoral studies at Princeton. Less than two years later, at the beginning of 1960, the American mathematician went to Rio de Janeiro. In 1961, Thom also visited Rio and, after becoming familiar with similar problems from singularity theory, Peixoto reformulated his result in his paper of 1962,28 using the term “generic” for the first time in the context of dynamical systems theory.
Peixoto starts with the following assertion: the fact that structurally stable systems form an open and dense subset of the space of all systems means they are “generic.” It is something significant “because these systems are precisely the ones that exhibit the simplest possible features being amenable to classification.”29 Thus, Peixoto manages to show that structurally stable systems defined on a compact bidimensional orientable surface are generic. But how are these systems precisely the ones that are classifiable?
We know the special features of structurally stable two-dimensional systems, given in conditions (1) and (2) of Andronov and Pontryagin. If these systems are generic, we can hope to classify them by their topologically distinct types, for instance, by their number of distinct singularities, even if it can be a difficult task.30 That is why we can say (p. 308) that the goal of demonstrating that structurally stable systems are generic is to establish conditions that make obtaining a classification result possible. Thus we propose calling it a pre-classification program.
But we remind ourselves that this problem was solved only for dynamical systems defined on specific two-dimensional spaces, and that in general the models dealt with have more dimensions. What happens in larger dimensions?
In the history of dynamical systems theory, in order to get to a pre-classification result for larger dimensions, the notions of stability and genericity had to be redefined using different mathematical tools. First of all, it was necessary to extend the definition of structurally stable systems for any dimension, which was done by Peixoto in the same year (1959) using the language of functional spaces.31
On the basis of this new definition, mathematicians need to find ways of describing as explicitly as possible those systems which are structurally stable in any dimension. In other words, mathematicians need to generalize conditions (1) and (2) of the characterization of Andronov and Pontryagin.
The work of the American mathematician Steve Smale proved to be of fundamental importance for providing a solution to the problem. As we have said, he met Peixoto in 1958 and the encounter, according to Smale, “sparked my interest in structural stability.”32 That is why he went to Rio de Janeiro to visit IMPA in 1960.
The generalization of the conditions characterizing structurally stable dynamical systems for dimensions greater than two is inspired by the work of Morse, which attests to the influence of the singularity theory of differentiable mappings. Morse was responsible for associating the number of singularities of a function to the topology of the manifold on which the function is defined. This motivated Smale to obtain a similar result in 1960, as he related the number of periodic trajectories of a dynamical system to the homology of the manifold on which the system is defined.33 Inspired by this result, he proposed, in the same article, to study systems with a finite number of periodic trajectories, and in which all trajectories approach periodic trajectories. As a consequence, these periodic trajectories are the only attractors of the system. In addition, he showed the stable and unstable (p. 309)
trajectories associated with those periodic trajectories must intercept transversally.34 Systems with these characteristics generalize the Andronov and Pontryagin conditions for higher dimensions and were called “Morse–Smale systems” by René Thom.
What we have here is the prototype for classifiable systems in higher dimensions that has been searched. The finiteness of the number of periodic trajectories satisfies the demand that attractors must be easily described for the purpose of classification. The demand that unstable and stable trajectories must intercept transversally35 is fundamental to establish robustness and genericity, something already suggested by Thom’s famous transversality theorem.36 It is not difficult to imagine that non-transversal intersections can be easily undone by very small perturbations. So, the qualitative aspect of trajectories in the presence of an intersection of this kind changes considerably after small perturbations. In Fig. 10.2, we show, respectively, a transversal and a non-transversal intersection of two curves in Rn . We can note that even if one perturbs the curves in Fig. 10.2(a), they remain transversal, whereas, in Fig. 10.2(b), a small perturbation can turn the intersection into a transversal one.
Yet, what remained to be shown is that Morse–Smale systems are open and dense, and therefore generic in the space of systems. This is the essential part of a famous conjecture proposed by Smale in 1960. If it were true, the classification program would be fulfilled, because it actually suffices to classify almost all systems by their number of attractors. Morse–Smale systems are classifiable systems and, if they are generic, it is possible to classify dynamical systems, in the sense that: (1) a system characterized by Morse–Smale properties is robust (its qualitative essential features are not destroyed by small perturbations); (2) any arbitrary system can be approached by a Morse–Smale, and, therefore, they are generic (dense).
(p. 310) 10.4 The Smale conjecture proven wrong: hyperbolicity, homoclinic points, and “chaos”
A deeper inquiry into Morse–Smale systems showed that the first condition holds, that is, that they are robust. It has been proven by the Brazilian mathematician Jacob Palis.37 Unfortunately, they are not dense in the space of systems. Smale’s conjecture is therefore false. In fact, a glance at some previous works of Poincaré, Birkhoff, or Hadamard would suggest that systems with a finite number of periodic points cannot be dense in the space of systems. Smale admitted that he did not know these works, and that is why he ended up being in error. However, beyond this first disappointment, the subsequent developments provide a very good example of how a wrong conjecture can become productive in mathematical research.
In his work on celestial mechanics, Poincaré had already noticed the possibility of what he called “doubly asymptotical points” or “homoclinic points.”38 They would be periodic points of a three-dimensional system’s dynamics, whose vicinity contains other trajectories with a very complex behavior. Poincaré studied the behavior of these trajectories by means of their intersections with a two-dimensional section, which associates to one point the point of the next return of the trajectory to the section. Thus, the study of the original dynamics of this now called “Poincaré section” is reduced to an analysis of a two-dimensional map defined on the section. The intersection of a periodic trajectory with this section is a fixed point of this map, and the other trajectories in its neighborhood also intersect the section in points (not fixed).39 There is a special kind of fixed point, also studied by Poincaré, now called “hyperbolic,” for which the dynamics in its neighborhood can be characterized by means of a stable manifold and an unstable manifold. The stable manifold is the set of all points that approach the fixed point under iteration of the map (shown by the almost vertical line in Fig. 10.3),40 and the unstable manifold is the set of all points that approach the fixed point under iteration of the inverse map (shown by the almost horizontal line in Fig. 10.3). In two dimensions, a fixed point of this type is a saddle point (the point p in Fig. 10.3).
If we start with a small ball of initial points centered on a saddle point and iterate the map, the ball will be stretched and squashed along the line of the unstable manifold, and the opposite occurs along the line of the stable manifold. We have a homoclinic point if the stable and the unstable invariant manifolds, from this same fixed point, intersect again. (p. 311)
On the Poincaré section, it gives rise to a very complex figure of trajectories. Figure 10.4 shows the successive intersections between the stable and the unstable manifolds.
Birkhoff had shown that in the neighborhood of a homoclinic point there are infinitely many others.41 The dynamic associated with a homoclinic point is not easy to destroy, as Norman Levinson already knew in 1949.42 Smale tells how he received a letter from him suggesting that the conjecture on the genericity of systems with a finite number of periodic points should be false.43 If Morse–Smale systems were dense, in a neighborhood of any system it would be possible to find a Morse–Smale system, which does not happen in the neighborhood of a system containing a homoclinic point. Indeed, in the presence of a homoclinic point it is possible to verify the phenomenon of infiniteness (p. 312)
of periodic points and to affirm its persistence (in particular, that this is a structurally stable phenomenon).
Smale easily admitted his error and immediately proposed a model to describe the set found by Levinson geometrically.44 This study gives birth to his “horseshoe” (in Fig. 10.5)45 which became very famous afterward as the prototype of chaotic behavior, associated mainly with its sensitivity to initial conditions.46
In a horseshoe, the cross section of the final structure corresponds to a Cantor set.47 This is one of its most interesting properties. Even if the complexity associated with a homoclinic intersection was already known, the model proposed by Smale allowed one to grasp the mechanism that produced its essential properties. In Section 10.5.1 we explore further the interesting phenomena with which these Cantor sets are related. (p. 313)
In an article called “Finding horseshoes in the beaches of Rio”—actually a note presented for the birthday of the Brazilian National Center of Scientific Development (CNPQ) in 1996—Smale asserts that the work on the horseshoe is the byproduct of the opportunity of being at that moment (in IMPA) at the confluence point of three traditions: the field in which Levinson (but also Cartwright-Littlewood) worked; the traditional contributions of Poincaré and Birkhoff; and the writings about structural stability of Soviet mathematicians that he had come to know from Peixoto.
In an inspiring example of how a wrong conjecture can produce novelty in mathematical practice, Smale uses the counter-example of the horseshoe in his favor and proposes to incorporate this phenomenon into a new dynamics prototype that should renew the classification on another basis. The role that periodic trajectories played in the Morse–Smale prototype should be played now by hyperbolic sets (of which the horseshoe is a particular case).48 If hyperbolic sets were proved to be structurally stable, they would be good candidates to constitute the new prototype of dynamics, which received the unfortunate denomination of “Axiom A.”49
(p. 314) In 1967, Smale published “Differentiable dynamical systems,” a long paper in which he summarized the advances of the theory up to that moment.50 At that point, the mathematical definitions needed to be adapted to the new classification program, and the concept of genericity itself underwent a transformation. In the first version of the conjecture, a certain kind of system was generic if it constituted an open and dense subset in the space of systems. Smale next proposes that the name “generic” must be associated with a property verified for a residual subset of systems. A subset of the space of systems is residual, in the Baire sense, if it is the intersection of a denumerable family of open and dense sets. Since, in the spaces here considered, a residual set is also dense, this new definition does not imply, from our point of view, a significant change in the conception of genericity.
Smale also suggested that it suffices to characterize the subsets of trajectories whose dynamics do not lead too far from their initial state (subsets of trajectories that do not escape). The subsets possessing this weak kind of recurrence are called “non-wandering.” They are all that mathematicians needed in order to describe the “interesting properties” of dynamical systems, since, in this domain, they are always considering phenomena that recur in some sense, and not the transient ones. In the new classification prototype, the non-wandering sets must be hyperbolic.
One more thing Smale did not know at the time was that in the Soviet Union, mathematicians, like Dmitri Victorovich Anosov, a student of Pontryagin, were already working to understand the structure of hyperbolic sets, in particular the one deriving from Hadamard’s example of geodesics on surfaces of negative curvature, later called Anosov diffeomorphisms.51
The definition of a new kind of system, based on homoclinic intersections was very proficuous to the research, since it can be associated with the existence of infinitely many periodic points. The work Anosov did on geodesic flows on surfaces of negative curvature showed that these are hyperbolic. Furthermore, this is a persistent behavior, something that could have already contradicted the first conjecture on the genericity of Morse–Smale systems.
Poincaré had suggested, in his writings on celestial mechanics, that periodic trajectories were the only breach52 through which we can penetrate in the space of trajectories in higher dimensions, in particular the trajectories in three-dimensional space, which he was studying at that time.53 Smale’s first conjecture asserted that periodic trajectories were effectively a good breach to penetrate the space of trajectories of a system of any dimension, which are far from being describable. Any system should be well understood if it could be approximated by a system with a finite number of periodic sets that attract all trajectories. But this program failed, and the next attempt was to test whether hyperbolic systems could be the breaches needed to describe the great complexity of (p. 315) dynamics. If hyperbolicity were a generic property of dynamical systems, it would be a very useful notion for furnishing a global theory, since hyperbolic systems can be very well understood (to such an extent that they can even be said to be “integrable”).54 But, once again, this is not true.
During the seventies, many examples were given revealing the existence of entire domains of dynamics covered with non-hyperbolic phenomena, which led to perplexity in the mathematical community.55 At this point, we must mention the influence of computational research—Lorenz and Hénon attractors are good examples as they exhibit a “persistent” non-hyperbolic dynamics. Besides those persistent non-hyperbolic attractors, there are other kinds of phenomena such as duplication of periods, coexistence of infinite pits in which the dynamics tend to disappear, and so forth.56 These are all “persistent” non-hyperbolic phenomena and Jacob Palis named them the “dark realm of dynamics.”57 We will see in the next section what “persistent” means in this context.
The perplexity that struck the researchers’ community in the seventies, generating a number of incomplete and challenging works, converged with the fact that, about this same moment, Western scientists became acquainted with the works of the Soviet school, leading the research in a new direction. These discoveries raised two new questions:
1) Would it be possible to find phenomena capable of generating non-hyperbolicity?
2) Are topological notions appropriate? Perhaps these notions, used in theory up to this time—such as structural stability or density as the definition for genericity58—were not appropriate for describing the behavior of the majority of dynamical systems.
Considerable efforts were made to penetrate the complement of the hyperbolic world—the “dark realm” of dynamics. Afterward, the research in the field took the two questions mentioned above as a point of departure, following two main directions: 1) the study of bifurcations of homoclinic tangencies (described in Section 10.5.1); and 2) ergodic theory (see Section 10.5.2). The latter theory uses probabilistic tools in a general study of dynamical systems, following the suggestion of the Soviet mathematicians. In the next topic we will briefly mention the relation these problems have to the issue of genericity.
(p. 316) 10.5 The changing definitions of genericity in the attempts to describe the “dark realm of dynamics”
10.5.1 Bifurcation of homoclinic tangencies
Many examples exhibited during the sixties and the seventies suggested that complex non-hyperbolic dynamics arise in relation to an interesting phenomenon called the “unfolding of a homoclinic tangency.”
Transversal intersections between stable and unstable manifolds, associated with periodic points, became an essential property for ascertaining that a system is robust, because non-transversal intersections, as shown in Fig. 10.2(b), can be easily destroyed. We have seen that transversality is associated with hyperbolicity. So, it is not very difficult to imagine that, if we want to understand non-hyperbolic behavior, we must study those cases in which the homoclinic points are not transversal intersections between stable and unstable manifolds.
Suppose p is a periodic point, if in the homoclinic point q the stable and the unstable manifold do not intersect transversally but are tangents to each other, as in Fig. 10.6, we call this last point a “homoclinic tangency.”
In the neighborhood of a homoclinic intersection there must be infinitely more points of the same type, produced by the intersections of the stable and unstable manifolds in the vicinity of the first one. So, a homoclinic point is part of a family of points of the same kind. It had already been proven that in the hyperbolic case, when the homoclinic point is a transversal intersection, there is a horseshoe in the vicinity of this point. In his geometrical model, as we cited above, Smale had already showed that a horseshoe is the product of two Cantor sets.
In the universe of dynamical systems defined on two-dimensional manifolds, as mathematicians want to study the trajectories of dynamical systems in the neighborhood of one system containing a homoclinic tangency, they define a family of diffeomorphisms fμ such that for μ = 0 this tangency is exhibited. In Fig. 10.6, we can see a picture of what happens for f0. When this system is perturbed, the value of μ changes and the homoclinic
(p. 317) tangency is unfolded. The system thus obtained can maintain the homoclinic tangency or can change it into a transversal homoclinic intersection. We can do an exercise to imagine the two curves in Fig. 10.6 are being perturbed—they can thus intersect in the same way or even transversally. As a consequence, the qualitative nature of the system can undergo an important change when homoclinic tangencies are unfolded. That is why this is called a “bifurcation point.”
In this context, genericity has to do with the following question: are tangencies a persistent phenomenon or can they be easily destroyed by small perturbations? What happens when we perturb (or unfold) a homoclinic tangency: “how many” phenomena arising after the perturbation are hyperbolic and “how many” are non-hyperbolic?
It was through the seminal works of Sheldon Newhouse59 that hyperbolicity was shown to be not dense in the space of a special kind of dynamical systems.60 The underlying mechanism here was the presence of a homoclinic tangency leading to the phenomenon called nowadays “Newhouse phenomena,” that is, non-negligible subsets of dynamical systems displaying infinitely many periodic attractors.
During the 1980s, mathematicians already knew some examples of non-hyperbolic behavior that functioned as prototypes.61 Some of them expected that a non-hyperbolic system containing a prototype could be approximated by another system containing a homoclinic tangency. If that happens, it would be possible to say that the unfolding of the tangency would give birth to the non-hyperbolic phenomena. There were many works of Jacob Palis, Floris Takens, Carlos Gustavo Moreira, Jean-Christophe Yoccoz, Sheldon Newhouse, and others in this direction.62 They tried to understand the appearance of complex non-hyperbolic behavior after the bifurcation of homoclinic tangencies.
Roughly speaking, mathematicians working in this field try to determine, in the space of parameters μ, of the dynamical systems fμ that are being unfolded, the size of the subsets that corresponds to transversal or tangent intersections. This is the main concern in the bifurcation theory of homoclinic tangencies: to understand complex phenomena in the complement of the hyperbolic world by means of homoclinic bifurcations.
Here, once again, we can see an attempt to draw a general picture of the major non-hyperbolic phenomena that can appear in dynamics. Yet, the goal is no longer to “classify” them, but to discover the phenomenon that triggers non-hyperbolicity. A system with a homoclinic tangency would indicate how to penetrate into the dark realm of dynamics.
In order to accomplish this project, it was necessary to describe the prevailing phenomena in dynamics after unfolding a homoclinic tangency. We have said earlier that non-hyperbolic phenomena are “persistent,” but this notion had to be defined. Mathematicians thus proposed mathematical refinements related to the definition of genericity in order to express the complexities of this new universe. In their research, they used some numerical invariants of Cantor sets to understand exactly what “persistence” means.
(p. 318) In this paragraph and the one that follows, we try to describe, as briefly and intuitively as possible, the mathematics involved in this direction of research. Suppose we have a family of dynamical systems defined by fμ such that for μ = 0 the system exhibits a homoclinic tangency. We want to know, near the bifurcation point .., how big in the parameter space (in the space of μ) is the set of values that correspond to hyperbolic or non-hyperbolic dynamical systems. The answer to this question has to do with different types of Cantor sets appearing in the space of parameters μ.
A system containing a tangency corresponds to a parameter μ0 belonging to a Cantor set. Since a Cantor set is full of holes, we could imagine that, with a small perturbation of the dynamics, it is possible to make the tangency disappear. But this is not the case. In order to study the unfolding of homoclinic tangencies, mathematicians had to impose conditions on Cantor sets that concern some invariants describing whether a set of this type is “small” or “large.” So, the possibility of destroying a tangency depends on some numerical properties that make one type of Cantor set different from another. Even if a Cantor set has no interior points, it is possible to define a property of “thickness,”63 that determines how it behaves with respect to perturbations. When a system is perturbed, the homoclinic intersections may or may not be persistent and the exact meaning of “persistence” here refers to the previously mentioned numeral properties of Cantor sets.
A notion of genericity emerges combining the notion of denseness—already in use in the hyperbolic context—with other notions concerning dynamically defined properties of Cantor sets. The main proposition in the theory asserts that if there is a homoclinic tangency between the stable and the unstable manifold of some periodic point and the related Cantor sets are sufficiently thick, the tangencies are persistent.64 In some sense, this result attests that when thickness is big enough, hyperbolicity cannot be generic.
Hopefully the brief explanations furnished thus far indicate how the concept of genericity necessarily changes in the development of research. As in the topological phase of dynamical systems theory, prevailing in the West up to the 1960s, the persistence of tangencies is mathematically expressed in terms of denseness. Nevertheless, denseness is studied by the authors cited in this section in relation with invariant numerical measures of Cantor sets.
10.5.2 Ergodic theory
Ergodic theory develops a probabilistic (i.e., measure-theoretic) approach in the study of determinist dynamical systems. Soviet mathematicians, such as Andrei Kolmogorov, had already introduced the idea of measure to describe the genericity of some kinds of (p. 319) systems. In the context of systems we are treating here, hyperbolic systems, the theory was proposed by Yacov Sinai, David Ruelle, and Robert Bowen. If we have a measure invariant by the dynamics, it is possible to describe “typical” properties of systems in the sense of the measure. This means that we do not describe the behavior of all the trajectories, but only that of “almost all” trajectories in a subset of trajectories with positive measure. What kind of properties can be studied in this way?
In the study of dynamical systems, sensitivity to initial conditions became a key notion, meaning that two motions departing from very close initial conditions can considerably diverge in the future. The systems presenting this behavior were called “chaotic.” In order to understand the behavior of trajectories, mathematicians first divide the space into cells and then study how often the trajectories pass into each cell (this implies that we do not consider individual trajectories anymore and that we do not need to know exactly where a trajectory is inside a cell). It is thus possible to analyze the dynamics by taking averages that constitute a probabilistic evaluation of deterministic dynamics.
The reason for the appearance of statistical laws is the instability of trajectories. There is such a variety of trajectories’ behaviors that the average aspect of trajectories in general tends to be stable.65 As an example of a general characteristic, there is the fraction of time that a trajectory spends in a given cell. The number of cells can get larger and we become interested in the time averages of the trajectories in each cell. This is a good example of a relevant statistical property that can be studied in deterministic systems.
Based on this property, it is possible to redefine the concept of stability in statistical terms. The topological definition of structural stability points out the qualitative (or topological) nature of the set of trajectories in a dynamical system. But in actual experiments, each time we observe the state of a system it is actually a different system that is being considered. Explained mathematically, suppose we have a system defined by means of a function f . When we study the iterations of xn, this leads to an xn+1 which is not exactly f(xn), because there is an aleatory noise. Thus after each iteration, we are considering a slightly different system.
We say that a system is stochastically stable if time averages are not affected by this noise. The notion of topological equivalence employed in the definition of structural stability gives too much importance to the structure of trajectories and, consequently, to their pathologies. From the viewpoint of stochastic research, structural stability does not eliminate a satisfactory number of pathologies and takes into account properties that are not sufficiently relevant. Furthermore, it is a typical property of hyperbolic systems. In order to study non-hyperbolic systems, it can be more appropriate to use a stochastic definition of stability.
Once stochastic stability is defined, the possibility of classification depends on the genericity of well-characterized stochastically stable systems. This characterization can be obtained with a defined system having a finite number of “good” attractors whose attraction basin covers “almost all” the ambient space, what means that the basin’s66 measure must be equal to the measure of the ambient space. Systems of this kind are (p. 320) significant in the description of the global aspect of dynamics since attractors characterize the asymptotical behavior of the system.
“Good” attractors are those with good measures, that is, physically relevant measures, and whose dynamics in attraction basins are stochastically stable. These can be non-hyperbolical attractors, but if they are finite, it is possible to classify dynamics by the number and nature of those attractors. Yet, this possibility relies on the genericity of such well-characterized stochastically stable systems.
Again, in the same style as the pre-classification program set by Peixoto and Smale, mathematicians in the 1980s proposed a conjecture regarding the genericity of stochastically stable systems. We note the central role that the research of IMPA, in Rio de Janeiro, played in this proposal, Jacob Palis being is a key figure in the general shape it acquired.67 The meaning of genericity employed here is still translated by the denseness of stable systems, since it is hard to define a measure in the infinite dimensional space of all systems.
At the International Mathematical Congress of 1954, held in Amsterdam, Kolmogorov gave a closing lecture with the title “The general theory of dynamical systems and classical mechanics.”68 This event played an important role in the development of what is now called Kolmogorov–Arnold–Moser (or KAM) theory. In his lecture, Kolmogorov discusses the occurrence of some special kinds of motions in the space of trajectories of Hamiltonian dynamical systems. He proposes to investigate the persistence of properties that are known for integrable systems after small, non-integrable, perturbations of the equations describing the system. But once again, what does persistence mean?
The issue at stake is to state that if certain conditions are fulfilled, after a sufficiently small perturbation of the integrable system, the space of motions is “mostly” filled by deformations of the invariant tori whose existence is known for the unperturbed system. “Mostly” here means that the complement of the set where it occurs has small measure.69 Starting with Kolmogorov’s version, the theorem was improved upon by V.I. Arnold and J. Moser in the 1960s, producing the result known nowadays as KAM theory.
It is not within the limits of our proposal to analyze the history or the implications of Kolmogorov’s theorem. We mention it only to show that the notion of “mostly” here becomes mathematically precise when a measure concept is employed.70 A phenomenon is persistent if it is frequent, that is, when the set of systems in which it is displayed has a big measure in the set of all systems.
In Section 10.1, we explained how the description of “almost all” systems in the universe of systems was translated in a topological language. In the study of bifurcations of homoclinic tangencies, some measure concepts were introduced. But before Western mathematics had gotten acquainted with the works developed in the Soviet Union, (p. 321) mathematicians such as Kolmogorov and Sinai had already suggested the inconvenience of topological notions to express the mathematical meaning of “almost all.”
The complement of a set of positive Lebesgue measure can be an open and dense set. Thus, a property that is topologically generic, fulfilled by an open and dense subset of systems, can be absent of all elements in a set of positive measure,71 which indicates that it is not generic in the measure-theoretic sense. So, a generic property should be defined as a property holding for a subset of “big” measure in the set of systems. A notion of genericity based on measures would be stronger than the notion based on denseness that is also used in the general study of dynamical systems. The only reservation with respect to this proposal is that it is hard to define an observable measure (a measure having a physical sense, such as Lebesgue measure) in the infinite dimensional space of all systems.
In the context of ergodic theory, generic properties must be related to statistical and probabilistic notions. But sometimes, it is necessary to use denseness as a definition for genericity, and in these cases, the mathematical tools with which genericity is assessed differ from those used to express the properties whose genericity is investigated.
10.6 Concluding remarks
We gave a very brief and panoramic report of different branches in dynamical systems theory, with special emphasis on the transformation undergone by key concepts, such as genericity. Our main goal was to furnish evidence for the thesis that mathematics very often develops under a tension in the search for a good compromise between a general understanding of mathematical objects and the properties making these objects relevant.
In the beginning, the definition of genericity was related to classification purposes, since not all systems are classifiable, and those that are classifiable have to be generic. The hitch is that both the concepts of “generic” and “classifiable” have to be defined. In an initial phase, a property was called generic if the subset of systems satisfying this property was dense in the space of all systems of a kind. As to the characterization of classifiable systems, it underwent quite a few changes. The topological aspect of the set of trajectories used by Peixoto and Smale, inspired by Andronov and Pontryagin’s works, was too restrictive and it was only appropriate for systems defined on two-dimensional manifolds. Even structural stability revealed itself as a property of hyperbolic systems, which are not very general themselves, even if they are very well understood. Bifurcation theory strived for an analysis of the mechanism that triggers non-hyperbolicity, noting that general results depend on numerical invariants of Cantor sets—some of which already existed and others had to be defined to advance the theory.
(p. 322) Finally, topological characterization turned out to be too restrictive, and only appropriate for the particular case of hyperbolic systems. Properties of a different nature, observed in the same systems, had to be expressed by statistical tools. From these tools, new stochastic definitions were proposed, which proved to be more useful for a general description of all systems, including non-hyperbolic systems. In turn, the advent of this new, probabilistic, notion of genericity triggered the search for new characterizations of what can be considered a good definition for “almost all” systems. From a certain perspective, the denseness definition is not as good as other ones using measure concepts. But, on the other hand, it is difficult to define a measure on the set mathematicians were working with.
Is there a definite answer to what is the best notion of genericity? That is not our point. As the complexity of the possible set of trajectories defined by a dynamical system increases, new notions are invented to deal with unexpected behaviors. When the research turned from bidimensional to more general systems, the hope of classifying all systems vanished, but the search for a general theory still motivated mathematicians. The stages of the development of the theory analyzed here furnish a good example of how the compromise between the generality of the description and the relevance of the studied properties entails a very productive tension in actual mathematical research.
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The Oxford Handbook of Generality in Mathematics and the Sciences. First Edition. Karine Chemla, Renaud Chorlay and David Rabouin. © Oxford University Press 2016. Publishing in 2016 by Oxford University Press.
(6) For the reader who is not familiar with these ideas, a geometric example can be given (in the spirit of F. Klein’s Erlangen program). In elementary plane geometry, one can consider three different groups, the group of translations (T), the group of direct isometries (I), and the group of similitudes (S). The second one is larger than the first one, since it also contains rotations; S is larger than I since it also contains homotheties (scalings). If only T is taken into account, plane geometric objects fall into a multitude of inequivalent classes; for instance, not all (infinite, straight) lines are equivalent, two lines being equivalent (up to a translation) if and only if they are parallel. If I is taken into account, all lines become equivalent and make up a single class; parallelism stops making sense. Yet I is enough to classify more complex geometric objects: for instance, two circles are equivalent relative to I if and only if they have the same radius (the “radius” thus forming a complete set of invariants). Moving on to S, all circles become equivalent and the notion of radius becomes meaningless. In Euclidean geometry (school variety), S is the standard group: the theorems proven for circles of radius 1 hold for circles of radius 2 all the same, since in the wording of theorems only ratios of lengths are referred to.
(7) This means that few mathematicians explored dynamical systems with the general perspective discussed here. Roque (2015) shows that a number of astronomers and mathematicians of Poincaré’s time employed some of the tools he proposed, as periodic solutions.
(11) With the Hessian being non-zero at each.
(12) A point p in a manifold M is a critical point if there is a local coordinate system (x 1 ,…,x n) about p such that ▽f(p) = 0. Such a critical point is non-degenerate if and only if the n × n matrix of second partial derivatives, called the Hessian of f at p, is non-singular, that is, its determinant does not vanish.
(16) “Une propriété (P) des applications f: Rn → Rp , définie localement en tout point de l’espace source sera dite générique, si l’ensemble des f qui ne présentent pas la propriété (P) en au moins un point d’un compact K de Rn forme dans L(RnRp : r) un ensemble fermé rare (fermé sans point intérieur)” (Thom, 1956b: 52). See also Thom (1956c).
(17) “Pour tout couple d’entiers n, p [concernant les applications de Rn dans Rp] (…) on se propose de décrire et classifier un certain ensemble de singularités (S), tel que les applications qui présentent des singularités du type (S) et uniquement de celles-là forment um ensemble ‘générique’ ” (Thom, 1956c: 59).
(27) In the paper of 1962 he extended this result to orientable two-dimensional manifolds that are compact and differentiable.
(30) Later on, Smale classified the topological distinct types of structurally stable systems on a two-dimensional compact differentiable manifold. He shows there are only a finite number of topologically distinct types of structurally stable systems having a given number of singularities and closed orbits.
Let X = (X 1, …,Xn),n ≥ 2 be a differential system of class C1 defined in the unit ball . The system X is said to be structurally stable in Bn if:
((i)) the vector field of X has no contact with the boundary S n−1 of Bn and, say, always points inward;
((ii)) there exists δ > 0 such that, whenever a system Y =( Y 1 ,…, Yn) satisfies
we can find a homeomorphism T of Bn onto itself mapping trajectories of X onto trajectories of Y.
(34) The stable and unstable trajectories are those neighboring the periodic one that respectively approach it or recede from it as time grows. The fact that they intercept transversally means this intersection occurs in a point with non-collinear tangents.
(35) The transversality property is satisfied only by these stable and unstable trajectories, which tend to periodic trajectories with increasing or decreasing time.
(38) These names and the description of the dynamics associated with this kind of systems were introduced, respectively, in Poincaré (1890) and Poincaré (1892–9). For a history of this discovery made by Poincaré, see Barrow-Green (1997) and Anderson (1994).
(40) This figure and the next one were taken from the article “Unstable periodic orbit” from the site Scholarpedia, at the address: http://www.scholarpedia.org/article/Unstable_periodic_orbits.
(45) I thank the Brazilian mathematician Maria José Pacifico for having offered me this beautiful image of a horseshoe.
(46) Alain Chenciner (2007) comments on how unfortunate is the designation of “chaos” in this case. As we will see later, the structure of trajectories in hyperbolic systems is very well understood, and has no reason to be called “chaotic.”
(47) Just to give an example, we explain how to obtain the Cantor ternary set. This set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third (1⁄3, 2⁄3) from the interval [0, 1], leaving two line segments: [0, 1⁄3] ∪ [2⁄3, 1]. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, 1⁄9] ∪ [2⁄9, 1⁄3] ∪ [2⁄3, 7⁄9] ∪ [8⁄9, 1]. This process is continued ad infinitum. The Cantor ternary set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process.
(48) Intuitively speaking, all points in a hyperbolic set are locally similar to a saddle point. For a more technical explanation, a compact smooth manifold M is considered and f: M → M is a diffeomorphism, with Df: TM → TM being the differential of f. An invariant subset of M is said to be hyperbolic if the tangent bundle in this subset admits a splitting into a sum of two subbundles, called the stable bundle and the unstable bundle. In the stable bundle we have a contraction and in the unstable bundle, an expansion. We refer here to systems for which the entire manifold M is hyperbolic.
(52) The term in French is “brèche.”
(56) Palis recalls the “supreme humiliation” of mathematicians in the seventies because the major part of those examples came from physics. The tendency of referees in mathematical journals, in this time, was to refuse articles from experimentalists that exhibited these kinds of behavior. Only Ruelle was more receptive (Palis, 1997).
(58) The Soviets realized this fact much earlier and used probabilistic notions instead, even in their study of hyperbolic systems. But their work started to have an influence on topological Western researchers by 1977, particularly during a meeting at Warsaw.
(60) However, let us point out that in the C1 topology it is still open.
(61) The Lorenz attractor, which characterized the famous “butterfly effect,” is one of these prototypes.
(63) The notion of thickness was introduced exactly to show that homoclinic tangencies are persistent. The precise meaning of this statement is that we can find a neighborhood U in the domain of dynamical systems treated here (C2 diffeomorphisms of a two-manifold) in a way that for each f ∈ U , there is a tangency between the stable and the unstable manifolds associated to a fixed point. Otherwise, this tangency can be obtained by a small perturbation of f (the diffeomorphisms which have a homoclinic tangency are dense in U). The persistence of homoclinic tangencies can be defined by this denseness property.
(66) For each attractor, its basin of attraction is the set of initial conditions leading to long-time behavior that approaches that attractor.
(69) The main question in KAM theory is to show that, generically, after small perturbations of integrable systems, the union of quasi-periodic tori has positive measure. The statement applies to many concrete models of classical mechanics, as in some versions of the n- body problem.
(70) We must recall that the measure concept is in straight relation with probability, one of the major fields of Kolmogorov’s research.
(71) A variant of the Cantor set furnishes an example of a nowhere dense set with positive measure. Remove from [0,1] all dyadic fractions of the form a/ 2nn in lowest terms for positive integers a and n and the intervals around them [a/ 2n − 1/22n+1, a/ 2n + 1/22n+1]; since for each n this removes intervals adding up to at most 1/2n+1, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1/2 (in fact just over 0.535… because of overlaps) and so in a sense represents the greatest part of the ambient space [0,1]. Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.