- 原书名：Dynamical Systems IX: Dynamical Systems With Hyperbolic Behaviour
- 原出版社： Springer
The book deals with smooth dynamical systems with hyperbolic behaviour of trajectories filling out "large subsets" of the phase space. Such systems lead to complicated motion (so-called"chaos"). The book begins with a discussion of the topological manifestations of uniform and total hyperbolicity: hyperbolic sets, Smale's Axiom A, structurally stable systems, Anosov systems, and hyperbolic attractors of dimension or codimension one. There are various modifications of hyperbolicity and in this connection the properties of Lorenz attractors, pseudo-analytic Thurston diffeomorphisms, and homogeneous flows with expanding and contracting foliations are investigated. These last two questions are discussed in the general context of the theory of homeomorphisms of surfaces and of homogeneous flows.
Chapter 1. Hyperbolic Sets
Chapter 2. Strange Attractors
Chapter 3. Cascades on Surfaces
Chapter 4. Dynamical Systems with Transitive Symmetry Group. Geometric and Statistical Properties
When at the beginning of the 60's the significance of hyperbolicity wasrecognized, investigations were first made of DS's in which such behaviour of a trajectory was expressed in the sharpest way--hyperbolicity is, so to speak,"total" (the hyperbolic character, noted above, of the change in time of the relative positions of two phase points, namely the original one and the slightly displaced one, holds under any direction of the displacement to one side of the original trajectory) and "uniform" (uniformity of the inequalities expressing the hyperbolicity with respect to the points, the displacements, and time).
The precise formulation of this version of hyperbolicity has led to the so-called hyperbolic sets, whose properties (both topological and metric (ergodic)) have aroused great interest. But in ergodic theory a lot of significant work hasbeen done since then, in connection with the weakening of the conditions of hyperbolicity in various directions. In the broadest terms, this progress is in connection with technical improvements rather than with new ideas of principle at the paradigmatic level. But however that may be, there have been considerable achievements in the metric theory occasioned by weakening the requirements of hyperbolicity. This has not been done in the topological theory (again speaking in the broadest terms). However, this does not mean that since the 60's there have been no significant achievements beyond the framework of total uniform hyperbolicity in the theory of smooth DS's, or that none of these achievements relates to DS's displaying some weaker form of hyperbolicity in their behaviour. Nevertheless, in the topological theory of DS's (so far?) no precise and workable notion of weakened hyperbolicity has been concocted. It would appear that here the gateway that might lead beyond the realm of total uniform hyperbolicity requires new ideas of principle. (At the same time, there remain many unsolved problems even within the bounds of these conditions.)
For all that, something now has emerged beyond the limits of the original version of hyperbolicity, although, as it seems to me, in the two most important cases, namely, DS's with a Lorenz attractor and certain cascades (DS's with discrete time) on surfaces, it is a question not so much of going beyond these limits, but rather of stretching them. (In essence, what have been weakened in these cases are not total and uniform hyperbolicity, but certain other "accompanying" conditions, so to speak. These are the conditions of continuity and smoothness (as ordinarily encountered in analysis); weakening them consists in violating them in some sense at certain points or on certain lines, and the condition that if a hyperbolic set of a flow contains equilibrium points, then they are isolated points of this set). In one way or another, these DS's are considered in the current volume. In this connection, certain information is also given on their metric properties, whereas questions of ergodic theory relating to hyperbolic sets are not touched upon in this volume; here we have nothing to add to the earlier results (given in (Bunimovich et al. 1985)).
The last two articles of this volume may outwardly appear to have no relation to hyperbolicity. In fact, the origin of their subject matter is only partially and fairly tenuously related to hyperbolicity. However in the development of this theme such connections have arisen and have turned out to be essential. These connections are a sufficient motivation for the inclusion of the last two articles in this volume although, of course, their contents deal with other matters. It is therefore worth saying a few words about these articles.
The investigation of cascades on closed surfaces is closely related to the classification of homeomorphisms of surfaces, more precisely, with the classification of isotopy classes of such homeomorphisms. The latter was the concern of topologists back in the 20's. One of the important questions here consisted in choosing "good" representatives of these isotopy classes; it must be recalled that successful representatives have usually turned out to be of great utility in many respects. Naturally, the representatives of some classes possess no by. perbolic properties. Until the 70's it was only such representatives that were known (leaving out the torus). Essential progress was achieved when "good'
representatives in the remaining isotopy classes had been successfully chosen. Leaving aside "mixed" (in technical jargon, reducible) cases, when the corresponding homeomorphisms behave differently in different parts of the surface, it can be said that the new representatives generate cascades with "typically hyperbolic" properties. Their difference from objects of ordinary hyperbolic theory is related not to the weakening of hyperbolicity conditions, but to the slight breakdown of smoothness at certain points. In this manner, progress in the theory of homeomorphisms of surfaces in the 70's was in no small degreestimulated by the development of the hyperbolic theory of DS's in the 60's. What has just been said is only one aspect of the theory of cascades on surfaces, this being the best known and most important. In the article included here, it is considered in the general context of this theory. ..
The article on certain DS's of algebraic origin, namely, homogeneous flows, occupies a special position in this volume. It is, to some extent, devoted to their ergodic properties. It might appear that this is close to Volume 2, but there is no such article there. The latter is partly accounted for by the fact that there was simply no room, due to the abundance of material in Volume 2. But the real reason is more one of principle.The investigation of the ergodic properties of homogeneous flows has its own specific character. Here an important role is played by the theory of Lie groups and their representations, while those ideas and methods forming the subject matter of Volume 2 recede into the background (although, of course, some of this is essentially used). In a number of cases homogeneous flows possess a certain amount of hyperbolicity (partial but uniform) and related to this are the corresponding geometric properties, as is the case throughout this volume. In essence, the algebra is then required for sorting out this geometry (without always mentioning this explicitly).
In connection with the discussion of the contents of this volume it is appropriate to mention three related sections of the theory of DS's that are not touched upon or only partly dealt with here. The first is certain questions of bifurcation theory where one has to deal with hyperbolicity. Information on this is contained in one of the earlier volumes of this edition (Arnol'd et al. 1986) (it is worth mentioning another new book (Wiggins 1988)); we have given only a few mentions of this. The second is "one-dimensional dynamics'', that is, the study of iterations of maps (in general, non-invertible) in a one-dimensional real or complex domain3 (in the latter case one talks about conformal dynamics whenever the iterated map is conformal). It arose independently of hyperbolic theory (and, if one is talking about conformal dynamics, somewhat earlier) but received an appreciable stimulus from the latter when it was realized (around 1970) that there was a "similarity" of behaviour of the trajectories between these two. There is also an influence the other way round: irreversible one-dimensional maps play an important auxiliary or heuristic role in the investigation of certain invertible DS's in higher dimensions that are peculiar to hyperbolic phenomena. It is only in this respect that these maps are referred to in the articles of the present volume. Even if such references had been discussed in greater detail, they would have reflected only part of one-dimensional dynamics and, of course, this arouses interest for other reasons. From a utilitarian point of view, it is simply a question that non-invertible one-dimensional maps arise in various questions of science and technology. Of course, invertible transformations relate more to the original ideas on DS's (as described at the beginning of (Anosov et al. 1985)), but there are nevertheless such problems that directly, or via some indirect route, lead to non-invertible one-dimensional maps.4 In its conceptual aspects, one-dimensional dynamics admits the almost unique possibility of a fairly full investigation of the complicated behaviour of DS's; that is, the behaviour both in the sense of the qualitative picture in phase space and in the sense of dependence on the parameters.5 In this connection it is worth mentioning the appropriate literature. In (Bunimovich et al. 1985) there is a small chapter on one-dimensional dynamics (primary attention being given to ergodic questions). In addition to the books and surveys referred to there, one can also mention (Eremenko and Lyubich 1989), (Lyubich 1986), (Sharkovskij et al. 1989), (Sharkovskij, Maistrenko and Romanenko 1986), (de Melo 1989), (Milnor 1990), (Nitecki 1982).
Another contiguous area left completely untouched in this book is hyperbolicity and bifurcations connected with it for infinite-dimensional systems. As is well known, reversion to such systems gives a satisfactory treatment of a number of problems for partial differential equations and for ordinary differential equations with a delay. It is natural that in this connection local questions (or questions of a similar character) should be developed in the first instance; but now hyperbolicity is also brought into play. Apparently, such
non-local entities have not been reflected at the textbook or survey-article level.
Finally, I should like to draw attention to four books of a relatively popular character that bear a relation to our theme. The books (Devaney 1989) and (Ruelle 1989) are introductions to a wide range of questions relating, in particular, to the complex behaviour of trajectories. Further, in such a situation there are so-called "fractal" sets which are quite unlike the usual geometric figures, being so violently "jagged" that it is reasonable to ascribe a fractional dimension to them (whence their name). The book (Falconer 1990) serves as an introduction to this topic. There is an album with coloured diagrams representing the results of numerical experiment giving rise to such objects (three quarters of these diagrams relate to conformal dynamics) and supplied with a certain amount of explanatory text (Pitgen and Richter 1986). The figures are very beautiful--in this respect they produce the same impression both for the specialist and for the complete outsider, mathematician or not.
To read this volume one needs general familiarity with the elements of the theory of DS's, not so much with regard to any advanced theorems, but rather with the general system of concepts, terminology, and so on. All this is contained in the article "Smooth dynamical systems", published in Volume 1 of the present series (Anosov et al. 1985) and to some extent precedes all the articles on DS's included in the other volumes. The requisite material from other branches of mathematics is in the main summarized in the preface to (Anosov et al. 1985), where one can also find the notation of a general mathematical character that is used (which is fairly standard). In certain parts additional material is required which is either recalled or (as the authors hope) is clear from the context. The chapter on homogeneous flows naturally takes up a special position in this respect. The reading of it requires not episodical, but constant (and sufficiently systematic) acquaintance with a number of questions that go beyond the above framework. All this is mentioned at the beginning of that chapter. ...
D. V. Anosov