# Welington de Melo's Books

We give some information about seven books by Welington de Melo. Each book is co-authored with one of the three mathematicians Jacob Palis, Sebastian van Strien and Edson de Faria.

Click on a link below to go to information on that book

Introdução aos sistemas dinâmicos (1978) with Jacob Palis

Geometric Theory of Dynamical Systems: An Introduction (1982) with Jacob Palis

One-dimensional dynamics (1993) with Sebastian van Strien

One dimensional dynamics: The mathematical tools (2001) with Edson de Faria

Mathematical aspects of quantum field theory (2007) with Edson de Faria

Mathematical tools for one-dimensional dynamics (2008) with Edson de Faria

Mathematical aspects of quantum field theory (2010) with Edson de Faria

1. Introdução aos sistemas dinâmicos (1978), by Jacob Palis, and Welington de Melo.
1.1. Note.

An English translation by Anthony K Manning with title Geometric Theory of Dynamical Systems: An Introduction was published in 1982 and information about this translation appears below.
2. Geometric Theory of Dynamical Systems: An Introduction (1982), by Jacob Palis, and Welington de Melo.
2.1. Acknowledgements.

This book grew from courses and seminars taught at IMPA and several other institutions both in Brazil and abroad, a first text being prepared for the Xth Brazilian Mathematical Colloquium. With several additions, it later became a book in the Brazilian mathematical collection Projeto Euclides, published in Portuguese, A number of improvements were again made for the present translation.

We are most grateful to many colleagues and students who provided us with useful suggestions and, above all, encouragement for us to present these introductory ideas on Geometric Dynamics. We are particularly thankful to Paulo Sad and, especially to Alcides Lins Neto, for writing part of a first set of notes, and to Anthony Manning for the translation into English.

2.2 Introduction.

We present in this book a view of the Geometric Theory of Dynamical Systems, which is introductory and yet gives the reader an understanding of some of the basic ideas involved in two important topics: structural stability and genericity.

This theory has been considered by many mathematicians starting with Poincaré, Lyapunov and Birkhoff. In recent years some of its general aims were established and it experienced considerable development.

More than two decades passed between two important events : the work of Andronov and Pontryagin (1937) introducing the basic concept of structural stability and the articles of Peixoto (1958-1962) proving the density of stable vector fields on surfaces. It was then that Smale enriched the theory substantially by defining as a main objective the search for generic and stable properties and by obtaining results and proposing problems of great relevance in this context. In this same period Hartman and Grobman showed that local stability is a generic property. Soon after this Kupka and Smale successfully attacked the problem for periodic orbits.

We intend to give the reader the flavour of this theory by means of many examples and by the systematic proof of the Hartman-Grobman and the Stable Manifold Theorems (Chapter 2), the Kupka-Smale Theorem (Chapter 3) and Peixoto's Theorem (Chapter 4). Several of the proofs we give are simpler than the original ones and are open to important generalisations. In Chapter 4, we also discuss basic examples of stable diffeomorpbisms with infinitely many periodic orbits. We state general results on the structural stability of dynamical systems and make some brief comments on other topics, like bifurcation theory. In the Appendix to Chapter 4, we present the important concept of rotation number and apply it to describe a beautiful example of a flow due to Cherry.

Prerequisites for reading this book are only a basic course on Differential Equations and another on Differentiable Manifolds the most relevant results of which are summarised in Chapter 1. In Chapter 2 little more is required than topics in Linear Algebra and the Implicit Function Theorem and Contraction Mapping Theorem in Banach Spaces. Chapter 3 is the least elementary but certainly not the most difficult. There we make. systematic use of the Transversality Theorem. Formally Chapter 4 depends on Chapter 3 since we make use of the Kupka-Smale Theorem in the more elementary special case of two-dimensional surfaces.

Many relevant results and varied lines of research arise from the theorems proved here. A brief (and incomplete) account of these results is presented in the last part of the text. We hope that this book will give the reader an initial perspective of the theory and make it easier for him to approach the literature.

Rio de Janeiro, September 1981.
Jacob Palis, Jr., Welington De Melo

2.3. Review by: Russell B Walker.
Mathematical Reviews MR0669541 (84a:58004).

This graduate-level text is a beautiful treatment of the foundations of dynamical systems, a large and thriving area of modern mathematics. It is a translation of a book first published in Portuguese as part of the Brazilian mathematics collection, "Projeto Euclides". Though it is for the most part self-contained, an elementary background in differential equations and some knowledge of the basic concepts of differential topology (manifolds, diffeomorphism, tangent bundles, etc.) would smooth the way.

Particularly appreciated by the reviewer is the selection and ordering of material. As a preparatory text for students interested in doing research in dynamical systems this text is to be highly recommended.

2.4. Review by: Joel Robbin.
The American Mathematical Monthly 91 (7) (1984), 448-449.

[After giving many technical definition Robin writes the following.]

The work of Smale and his school in the sixties and seventies greatly advanced our understanding of differentiable dynamics ... . In particular, it emerged that structurally stable systems are (if suitably defined) generalisations of the map of the interval considered above and a wealth of examples and theories beyond structural stability were discovered. It also emerged that structurally stable systems are not as ubiquitous as first thought and much current research is devoted to the study of systems which are definitely not structurally stable.

The book under review provides a good introduction to this theory. It treats the $(C^{0})$ linearisation problem mentioned above but in n-dimensions. It also discusses the theory of structurally stable (continuous time) systems on 2-manifolds as developed by Peixoto. An indication of the more advanced theory is given, and there is a bibliography with 122 entries. The book contains ample detail, plenty of examples, and 62 figures illustrating the text. Both authors have themselves made important contributions to the subject, and it is to be hoped that their effort will attract new graduate students to a beautiful subject.

Bulletin mathématique de la Société des Sciences Mathématiques de la RépubliqueSocialiste de Roumanie 32 (80) (1) (1988), 92.

At the time we are writing these lines this remarkable book of the well-known Brazilia mathematicians Palis and de Melo is already a reference paper for specialists in particular-sciences applied mathematics and undergraduate students in mathematics interested into an easy but rigorous introduction to two main topics of the dynamical systems theory: structural stability and genericity. By now, the book is available in Portuguese, English and Russian. It contains four chapters; the first one summarises the most relevant results on differential equations and differentiable manifolds, Chapter 2 deals with local stability, Chapter 3 is devoted to the Kupka-Smale theorem and the last one is concerned with genericity and stability of Morse-Smale vector fields. To a wider accessibility many treated examples, comments orientative figures, remarks are included. In publishing this book Springer-Verlag accomplished a very useful job.
3. One-dimensional dynamics (1993), by Welington de Melo and Sebastian van Strien.
3.1. From the Publisher.

One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarise the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearisability of circle diffeomorphisms due to M Herman, J-C Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurston; also included are an exposition on Hofbauer's tower construction and a result on full multimodal families (this last result solves a question posed by J Milnor).

3.2. Summary.

An account of the state-of-the-art in one-dimensional dynamical systems. The subject is studied from a combinatorial, continuous, ergodic and smooth point of view. It includes coverage of new developments on universality and renormalisation.

3.3. Contents.

- 0. Introduction.

- I. Circle Diffeomorphisms.

- 1. The Combinatorial Theory of Poincare.
- 2. The Topological Theory of Denjoy.
- 2.a The Denjoy Inequality.
- 2.b Ergodicity.
- 3. Smooth Conjugacy Results.
- 4. Families of Circle Diffeomorphisms; Arnol'd tongues.
- 5. Counter-Examples to Smooth Linearizability.
- 6. Frequency of Smooth Linearizability in Families.
- 7. Some Historical Comments and Further Remarks.

- II. The Combinatorics of One-Dimensional Endomorphisms.

- 1. The Theorem of Sarkovskii.
- 2. Covering Maps of the Circle as Dynamical Systems.
- 3. The Kneading Theory and Combinatorial Equivalence.
- 3.a Examples.
- 3.b Hofbauer's Tower Construction.
- 4. Full Families and Realization of Maps.
- 5. Families of Maps and Renormalization.
- 6. Piecewise Monotone Maps can be Modelled by Polynomial Maps.
- 7. The Topological Entropy.
- 8. The Piecewise Linear Model.
- 9. Continuity of the Topological Entropy.
- 11. Some Historical Comments and Further Remarks.

- III. Structural Stability and Hyperbolicity.

- 1. The Dynamics of Rational Mappings.
- 2. Structural Stability and Hyperbolicity.
- 3. Hyperbolicity in Maps with Negative Schwarzian Derivative.
- 4. The Structure of the Non-Wandering Set.
- 5. Hyperbolicity in Smooth Maps.
- 6. Misiurewicz Maps are Almost Hyperbolic.
- 7. Some Further Remarks and Open Questions.

- IV. The Structure of Smooth Maps.

- 1. The Cross-Ratio: the Minimum and Koebe Principle.
- 1.a Some Facts about the Schwarzian Derivative.
- 2. Distortion of Cross-Ratios.
- 2.a The Zygmund Conditions.
- 3. Koebe Principles on Iterates.
- 4. Some Simplifications and the Induction Assumption.
- 5. The Pullback of Space: the Koebe/Contraction Principle.
- 6. Disjointness of Orbits of Intervals.
- 7. Wandering Intervals Accumulate on Turning Points.
- 8. Topological Properties of a Unimodal Pullback.
- 9. The Non-Existence of Wandering Intervals.
- 10. Finiteness of Attractors.
- 11. Some Further Remarks and Open Questions.

- V. Ergodic Properties and Invariant Measures.

- 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.
- 2. Invariant Measures for Markov Maps.
- 3. Constructing Invariant Measures by Inducing.
- 4. Constructing Invariant Measures by Pulling Back.
- 5. Transitive Maps Without Finite Continuous Measures.
- 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson's Theorem.
- 7. Some Further Remarks and Open Questions.

- VI. Renormalization.

- 1. The Renormalization Operator.
- 2. The Real Bounds.
- 3. Bounded Geometry.
- 4. The PullBack Argument.
- 5. The Complex Bounds.
- 6. Riemann Surface Laminations.
- 7. The Almost Geodesic Principle.
- 8. Renormalization is Contracting.
- 9. Universality of the Attracting Cantor Set.
- 10. Some Further Remarks and Open Questions.

- VII. Appendix.

- 1. Some Terminology in Dynamical Systems.
- 2. Some Background in Topology.
- 3. Some Results from Analysis and Measure Theory.
- 4. Some Results from Ergodic Theory.
- 5. Some Background in Complex Analysis.
- 6. Some Results from Functional Analysis.

3.4. From the Preface.

One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified account of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research.

Let us quickly summarise the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearisability of circle diffeomorphisms due to M Herman, J-C Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurston; also included are an exposition on Hofbauer's tower construction and a result on full multimodal families (this last result solves a question posed by J Milnor). In the third chapter we analyse structural stability and hyperbolic properties of one-dimensional systems with a simplified proof of a result of R Marie. This chapter has a section describing the ideas that are used in the theory of rational maps on the Riemann sphere corresponding to those which are used here; it also serves as an introduction to the last chapter. The fourth chapter shows that Denjoy's result for circle diffeomorphisms also holds for non-invertible maps and that the period of the attractors of these maps is bounded. The first part of this result extends work of J Guckenheimer, J-C Yoccoz, A M Blokh and M Yu Lyubich. The first three sections of Chapter IV give all the distortion tools necessary for the remaining chapters.

From Chapter V onwards, we confine ourselves to unimodal maps. The fifth chapter gives an account of the state-of-the-art concerning ergodicity, Cantor attractors and invariant measures. ... In the final chapter, a complete proof is given of D Sullivan's recent results on the universal structures of one-dimensional systems coming from renormalisation. This chapter uses ideas from complex analysis, in particular Teichmüller theory. The necessary background for this book is treated in an appendix.

This has resulted in a lengthy volume. However, the reader should not be discouraged by this. The book is organised in such a way that each of the chapters is essentially independent of the others. For this reason, we have defined some concepts twice or more. (For example the notion of wandering interval is defined in both Chapter I and II.)

We have used Chapter II of this text for an introductory course at first year graduate level on the combinatorial properties of piecewise monotone maps. Viviane Baladi used Chapter I and the first part of Chapter II for an 8 week introductory graduate course in dynamical systems. Sections IV.1-IV-3 and V.1-V.6 were used for another course on the metric theory of unimodal maps. A more advanced graduate course on the theory of renormalisations included Sections III.1, IV.1-IV.3 and Chapter VI. Most exercises include substantial hints (some of these hints are even complete proofs).

We should emphasise that we have not tried to give a complete account of all developments in one-dimensional dynamics. Most notably we have only given a short introduction to the combinatorial theory of one-dimensional systems ...We have not touched on thermodynamical theory, singularity spectra, decay of correlations and such matters, though some references to these subjects are given.

This book grew from notes written by the first author which were distributed during a course organised by the Brazilian Mathematical Society. The first part of this book retains the same structure as those notes but has been very much expanded. Moreover, the last two chapters are new. In these chapters results are given on ergodic properties of one-dimensional dynamical systems and also a complete account is given of Dennis Sullivan's proof of the renormalisation conjectures of Feigenbaum, Coullet and Tresser. Exercises have been added.

... We would like to thank Dennis Sullivan for many inspiring discussions and in particular spending many hours with us explaining his renormalisation results. The first author would like to thank IHES and the Technical University of Delft for the hospitality during several visits that were essential for the preparation of this book.

Amsterdam, Rio de Janeiro
Welington de Melo, Sebastian van Strien
December 1992

3.5. On Mathematical beauty.

As we hope to show in this book, the theory of one-dimensional dynamical systems has a very beautiful structure. It is built-up in several layers: the first one is based on the order structure on an interval or a circle and the Intermediate Value Theorem for continuous maps. This already leads to a surprisingly rich 'combinatorial' theory. The second layer is connected to the a ne structure and based on the fact that in one-dimensional space the size (i.e., volume) and the diameter of a connected set are the same: an example of a tool of this type is the Mean Value Theorem. Thirdly, one has the projective group acting on the real line which will lead to the Koebe Principle. These three structures allow the study of the orbit of a configuration of respectively two, three and four points. Finally, an interval or a circle can be embedded in the complex plane where one has the notion of conformal and quasiconformal maps. This leads to ways to define 'geodesic arcs' between such dynamical systems.

Because one has such a good mathematical framework, the general theory of one-dimensional dynamical systems is very well developed. For example, we can completely describe the dynamics which can occur in these systems in a topological sense. This dynamics can be extremely complicated (for example infinitely many periodic points can coexist in a far more complicated way than in the usual horseshoe maps) and yet is completely understood. In a metric sense, the orbit structure is also increasingly understood. Surprisingly, one often has rigidity (or universality as physicists choose to call it): the metric structure of orbits is completely determined by the topological one.

Moreover, the mathematical tools that are used in this theory come from different and beautiful branches of mathematics - number theory, topology, ergodic theory, complex analysis, real analysis, general dynamical systems and foliation theory to name a few.

3.6. Review by: Feliks Przytycki.
Mathematical Reviews MR1239171 (95a:58035).

This is a lengthy book, a survey and monograph on one-dimensional dynamics. It unifies the existing theory, and contains several simplifications and precise expositions of very deep recent results. It proceeds from the beginning to the frontiers of research.

3.7. Review by: K M Brucks.
SIAM Review 37 (3) (1995), 461-462.

In recent years there has been a burst of interest in one-dimensional dynamics. Previously, many of the results in one-dimensional dynamics had been difficult to access via the literature and/or had been part of the "folk knowledge." This is no longer the case. The book under review is a must for anyone doing research in the area of one-dimensional dynamics. As the authors point out in their introduction, questions within one-dimensional dynamics fall into four categories: combinatorial, topological, ergodic, and smooth. The book under review provides, in a very readable manner, a wealth of information in all four categories.
...
This reviewer has successfully used Chapter two in both graduate-level dynamics courses and seminars. Exercises are provided throughout the book and lots of historical information is given in each chapter. Students found the material challenging yet beautiful. The authors provide an Appendix with lots of helpful background material. The bibliography is rich with references!

Again, this book is an absolute must for anyone working with one-dimensional dynamics! The book is a beautiful and extremely useful collection of mathematics. ... This reviewer highly and warmly recommends this book! Last, two other recent texts in one-dimensional dynamics are also excellent books. In [L Alseda, J Llibre and M Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One (1993)] a treatment of both combinatorial and topological dynamics is presented; while the main emphasis in [L S Block and W A Coppell, Dynamics in one dimension (1992)] is topological dynamics. Those working in dynamics and, in particular, one-dimensional dynamics, are fortunate to have such beautiful and useful texts available.
4. One dimensional dynamics: The mathematical tools (2001), by Edson de Faria and Welington de Melo.
4.1. Review by: Saeed Zakeri.
Mathematical Reviews MR1851555 (2002f:37077).

This short book introduces the reader to the mathematical tools which play an essential role in the modern theory of holomorphic dynamical systems. The main emphasis is on basic techniques from geometric complex analysis and quasiconformal theory in dimension one. The significance of these techniques is often illustrated by their dynamical applications.
...
Much of the material presented in the book is classical and the dynamical applications are rather well-known. Nevertheless, the authors deserve credit for successfully giving in 100 pages a panorama of the mathematical apparatus used in holomorphic dynamics. This is certainly a great service to those who want to get acquainted with the subject, since it saves them from having to leaf through hundreds of pages of the fast-growing literature. Moreover, the authors have shown great discernment in selecting the results and finding the most elegant proofs available for them. The outcome of their effort is not a textbook in quasiconformal theory or holomorphic dynamics; it is an excellent expository work that can be read and enjoyed by students and experts alike.
5. Mathematical aspects of quantum field theory (2007), by Edson de Faria and Welington de Melo.
5.1. Contents.

1. Classical mechanics;
2. Quantum mechanics;
3. Relativity, the Lorentz group and Dirac's equation;
4. Fiber bundles, connections and representations;
5. Classical field theory;
6. Quantization of classical fields;
7. Perturbative quantum field theory;
8. Renormalization;
9. Standard model;
Bibliography.

5.2. From the preface.

In this book we attempt to present some of the main ideas of quantum field theory for a mathematical audience. As mathematicians, we feel deeply impressed - and at times quite overwhelmed - by the enormous breadth and scope of this beautiful and most successful of physical theories.

Through the centuries, mathematics has always provided physics with a variety of tools, oftentimes on demand, for the solution of fundamental physical problems. But the past century has witnessed a new trend in the opposite direction: the strong impact of physical ideas not only in the formulation, but in the very solution to mathematical problems. Some of the most well-known examples of such an impact are: (1) the use of renormalisation ideas by Feigenbaum, Coullet and Tresser in the study of universality phenomena in one-dimensional dynamics; (2) the use of classical Yang-Mills theory by Donaldson to devise invariants for four-dimensional manifolds; (3) the use of quantum Yang-Mills by Seiberg and Witten in the construction of new invariants for 4-manifolds; (4) the use of quantum theory in three dimensions leading to the Jones-Witten and Vassiliev invariants. There are several other examples.

Despite the great importance of these physical ideas, mostly coming from quantum theory, they remain utterly unfamiliar to most mathematicians. This we find quite sad. As mathematicians, we found it very difficult while researching this book to absorb physical ideas, not only because of the eventual lack of rigour - this is rarely a priority for physicists - but primarily due to the absence of clear definitions and statements of the concepts involved.

5.3. Review by: John Palmer.
SIAM Review 54 (1) (2012), 199-203.

Some years ago at a conference in Chicago on representation theory, attended by mathematicians and physicists, the mathematicians in the audience asked the lecturing physicist what the symbol $\phi$ he used so liberally signified. He turned to look at the blackboard and said, "Why, $\phi$ is a boson!" A little later in the conference when the earnest question, "Is a quantum group a group?" was answered by "It depends on what you mean by a group," the answer provoked a storm of protest from the mathematicians who were in no doubt about what constituted a group. Now this was a conference about mathematical results in representation theory that were of mutual interest to mathematicians and physicists, but the different styles in which the two groups work makes communication difficult. In the twentieth century, mathematics made tremendous advances in algebra, topology, geometry, and analysis largely by insisting on careful definitions and axiomatic formulations. The revolution in physics that emerged from a new understanding of space and time and the subsequent geometrisation of gravitation seeded an intense development of Riemannian and pseudo-Riemannian geometry in mathematics. The later quantum revolution in physics very quickly gave birth to the spectral theory of operators in Hilbert space and catalysed interest in group representations in the realm of mathematics. If either one of these revolutions had happened without the other, the schism that separates mathematics from physics today would be much less severe. However, the synthesis of these two revolutions (in 4 space-time dimensions) has proved to be beyond the scope of the mathematics developed to this day, and in the case of gravity and quantum mechanics beyond the scope of physics as well. The book Mathematical Aspects of Quantum Field Theory by Edeson de Faria and Wellington de Melo is an attempt to explain the particular marriage of quantum mechanics with the linear geometry of Minkowski space-time that physicists know as the Standard Model from a point of view congenial to the mathematician. Mostly this involves working through the traditional physicist's approach to the subject (renormalised perturbation theory) with special care about definitions and the recognition that most of the continuum constructions in the physicist's approach don't make mathematical sense without extra effort. This is very succinctly accomplished in the space of 120 pages in a book of 300 pages. Much of the rest of the book is devoted to an account of the mathematics, like spectral theory, representation theory, and the geometry of connections that is a backdrop for quantum mechanics and the gauge fields of the Standard Model.
6. Mathematical tools for one-dimensional dynamics (2008), by Edson de Faria and Welington de Melo.
6.1. From the Preface.

Our main goal in this book is to introduce the reader to some of the most useful tools of modern one-dimensional dynamics. We do not aim at being comprehensive but prefer instead to locus our attention on certain key tools. We believe that the to pics covered here are representative of the depth and beauty of the ideas in the subject. For each tool presented in the book, we have selected at least one non-trivial dynamical application to go with it.

Almost all the topics discussed in the text have their source in complex function theory and the related areas of hyperbolic geometry, quasiconforrnal mappings and Teichmüller theory. This is true even of certain tools, such as the distortion of cross-ratios, that are applied to problems in real one-dimensional dynamics. The main tools include three deep theorems: the uniformisation theorem (for domains in the Riemann sphere), the measurable Riemann mapping theorem and the Bers-Royden theorem on holomorphic motions. These are presented, with complete proof, in Chapters 3, 4 and 5 respectively.

The present book originated in a set of notes for a short course we taught at the 23rd Brazilian Mathematics Colloquium (IMPA, 2001). We have benefited from useful criticism of the original notes from friends and colleagues, especially Andre de Carvalho, who read through them and found several inaccuracies. We are also grateful to two anonymous referees for their perspicacious remarks and suggestions.

The drawings of Julia sets and the Mandelbrot set found in this book were made with the help of computer programs written by C Mullen . ...We wish to thank also Dayse H Pastore for her help with the figures for the original colloquium notes, some of which appear in the present book.

Finally, it is quite an honour to see our book published by Cambridge University Press, and in such a prestigious series. We are grateful to David Tranah and Peter Thompson for this opportunity and for their patient and highly professional support.

Edson de Faria and Welington de Melo
Sao Paulo and Rio de Janeiro
January 2008

6.2. From the Introduction.

It is fair to say that the subject known today as complex dynamics - the study of iterations of analytic functions - originated in the pioneering works of P Fatou and G Julia early in the twentieth century. In possession of what was then a new tool, Montel's theorem on normal families, Fatou and Julia each investigated the iteration of rational maps of the Riemann sphere and found that these dynamical systems had an extremely rich orbit structure. They observed that each rational map produced a dichotomy of behaviour for points on the Riemann sphere. Some points - constituting a totally invariant open set known today as the Fatou set - showed an essentially dissipative or wandering character under iteration by the map. The remaining points formed a totally invariant compact set, today called the Julia set. The dynamics of a rational map on its Julia set showed a very complicated recurrent behaviour, with transitive orbits and a dense subset of periodic points. Since the Julia set seemed so difficult to analyse, Fatou turned his attention to its complement (the Fatou set). The components of the Fatou set are mapped to other components, and Fatou observed that these seemed to eventually to fall into a periodic cycle of components. Unable to prove this fact, but able to verify it for many examples, Fatou nevertheless conjectured that rational maps have no wandering domains. He also analysed the periodic components and was essentially able to classify them into finitely many types.

It soon became apparent that even the local dynamics of an analytic map was not well understood. It was not always possible to linearise the dynamics of a map near a fixed or periodic point, and remarkable examples to that effect were discovered by H Cremer. In the succeeding decades researchers in the subject turned to this linearisation problem, and the more global aspects of the dynamics of rational maps were all but forgot ten for about half a century.

With the arrival of fast computers and the first pictures of the Mandelbrot set, interest in the subject began to be revived. People could now draw computer pictures of Julia sets that were not only of great beauty but also inspired new conjectures. But the real revolution in the subject came with the work of D Sullivan in the early 1980s. He was the first to realise that the Fatou-Julia theory was strongly linked to the theory of Kleinian groups, and he established a dictionary between the two theories. Borrowing a fundamental technique first used by Ahlfors in the theory of Kleinian groups, Sullivan proved Fatou's long-standing conjecture on wandering domains. With this theorem Sullivan started a new era in the theory of iterations of rational functions.

Our goal in this book is to present some of the main tools that are relevant to these developments (and to other more recent ones). Our efforts are concentrated on the exposition of only a few tools. We tried to select at least one interesting dynamical application for each tool presented, but it was not possible to be very systematic. Many interesting techniques had to be omitted, as well as many of the more interesting contemporary applications. There are a number of superbly written texts in complex dynamics with a more systematic exposition of theory; ...

The remainder of this introduction is devoted to a more careful explanation of the basic concepts involved in the above discussion and also to a brief description of the contents of the book. ...

6.3. Review by: Petra Bonfert-Taylor.
Mathematical Reviews MR2455301 (2010a:37090).

The purpose of this book is to introduce the reader to some of the major mathematical tools from the theory of complex dynamics. The book aims to be self-contained in that complete proofs of most of these tools are presented. Exercises at the end of each chapter supplement the material. For each major tool presented, at least one interesting dynamical application is introduced as well.

The book is directed at graduate students and researchers working in dynamical systems and related fields.
7. Mathematical aspects of quantum field theory (2010), by Edson de Faria and Welington de Melo.
7.1. From the Publisher.

Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and has inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantisation of classical fields, perturbative quantum field theory, renormalisation, and the standard model.

The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries, and group representations.

7.2. Foreword by Dennis Sullivan.

Mathematicians really understand what mathematics is. Theoretical physicists really understand what physics is. No matter how fruitful the interplay between the two subjects, the deep intersection of these two understandings seems to me to be quite modest. Of course, many theoretical physicists know a lot of mathematics. And many mathematicians know a fair amount of theoretical physics. This is very different from a deep understanding of the other subject. There is great advantage in the prospect of each camp increasing its appreciation of the other's goals, desires, methodology, and profound insights. I do not know how to really go about this in either case. However, the book in hand is a good first step for the mathematicians.

The method of the text is to explain the meaning of a large number of ideas in theoretical physics via the splendid medium of mathematical communication. This means that there are descriptions of objects in terms of the precise definitions of mathematics. There are clearly defined statements about these objects, expressed as mathematical theorems. Finally, there are logical step-by-step proofs of these statements based on earlier results or precise references. The mathematically sympathetic reader at the graduate level can study this work with pleasure and come away with comprehensible information about many concepts from theoretical physics ... quantisation, particle, path integral ... After closing the book, one has not arrived at the kind of understanding of physics referred to above; but then, maybe, armed with the information provided so elegantly by the authors, the process of infusion, assimilation, and deeper insight based on further rumination and study can begin.

Dennis Sullivan
East Setauket, New York

7.3. Extract from the Preface.

In this book, we attempt to present some of the main ideas of quantum field theory (QFT) for a mathematical audience. As mathematicians, we feel deeply impressed - and at times quite overwhelmed - by the enormous breadth and scope of this beautiful and most successful of physical theories.

For centuries, mathematics has provided physics with a variety of tools, oftentimes on demand, for the solution of fundamental physical problems. But the past century has witnessed a new trend in the opposite direction: the strong impact of physical ideas not only on the formulation, but on the very solution of mathematical problems. Some of the best-known examples of such impact are (1) the use of renormalization ideas by Feigenbaum, Coullet, and Tresser in the study of universality phenomena in one-dimensional dynamics; (2) the use of classical Yang–Mills theory by Donaldson to devise invariants for four- dimensional manifolds; (3) the use of quantum Yang-Mills by Seiberg and Witten in the construction of new invariants for 4-manifolds; and (4) the use of quantum theory in three dimensions leading to the Jones-Witten and Vassiliev invariants. There are several other examples.

Despite the great importance of these physical ideas, mostly coming from quantum theory, they remain utterly unfamiliar to most mathematicians. This we find quite sad. As mathematicians, while researching for this book, we found it very difficult to absorb physical ideas, not only because of eventual lack of rigour - this is rarely a priority for physicists - but primarily because of the absence of clear definitions and statements of the concepts involved. This book aims at patching some of these gaps of communication.
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The first version of this book was written as a set of lecture notes for a short course presented by the authors at the 26th Brazilian Math Colloquium in 2007. For this Cambridge edition, the book was completely revised, and a lot of new material was added.

7.4. Review by: Walter D van Suijlekom.
Mathematical Reviews MR2683277 (2012d:81227).

During the last century, the development of a quantum theory of fields was mostly done in parallel with experiments in high-energy physics, which actually was the main motivation for studying such infinite-dimensional quantum systems. It came as completely unexpected that ideas from quantum field theory (QFT) could be applied in pure mathematics, such as in the construction of topological invariants of manifolds in dimension three and four. Naturally, the physics literature on QFT is quite unfriendly for the pure mathematician. This book - which is written by two mathematicians - aims at a mathematical audience, presenting the basic ideas and tools from QFT with as much rigour as possible, whilst being honest when discussing ill-defined objects such as the path integral.

Given the size of the book one does not expect to become an expert on quantum field theory after reading it; this is of course not the intention of the authors. Nevertheless, as Dennis Sullivan puts it in his foreword: "armed with the information provided so elegantly by the authors, the process of infusion, assimilation, and deeper insight based on further rumination and study can begin."

Last Updated February 2023