A first course in dynamics with a panorama of recent developments.

*(English)*Zbl 1027.37001
Cambridge: Cambridge University Press. x, 424 p. (2003).

This is an excellent presentation of the modern theory of dynamical systems at the advanced undergraduate to intermediate graduate level written by two renowned experts in the field. As suggested by the title, the book is divided into two principal parts – the course introducing the subject and a more advanced panorama presenting further results on dynamical systems. There are numerous insertions of more advanced material in the text of the main course which allow instructors to choose the appropriate level of depth and to “spice up” the lectures whenever desired.

The textbook opens with a nice Introduction whose main purpose is to rouse the interest of the reader to this thrilling and fast developing branch of modern mathematics. From the very beginning, the reader is captivated by skillfully written historical comments and challenged with theoretical exercises and numerical experiments.

The course itself consists of seven chapters. The exposition starts with Chapter 2, where the simplest dynamics exhibited in systems with stable asymptotic behavior is examined. The topics include, but are not limited to, linear maps and linearization, nondecreasing maps of an interval and bifurcations, quadratic maps, fractals. Linear maps and differential equations in the plane and in higher dimension are the subject of the systematic study in Chapter 3. Recurrent behavior of orbits which eventually return close to initial states, but not the exact departure points, is studied in Chapter 4, where rotations of the circle, applications of density and uniform distribution, invertible circle maps, and Cantor phenomena are discussed. The study of recurrence and equidistribution is continued in Chapter 5 for the case of higher dimension. Chapter 6 deals with important aspects of the conservative systems: preservation of the volume and recurrence, Newtonian systems of classical mechanics, and billiards. Simple systems with complicated orbit structure are examined in Chapter 7, where numerous important properties and examples are addressed with a special attention paid to topological transversality, chaos, coding, independence, entropy, and mixing. The discussion of entropy and chaos continues in Chapter 8 which concludes the first part of the book.

The second part of the book opens with Chapter 9 which establishes the link between both parts by emphasizing once again the crucial role played by simple dynamics in understanding more complicated behaviors. The results collected in this chapter are important for the study of dynamical systems: the implicit function theorem, the inverse function theorem, and the smooth contraction principle are among these. Fundamental theorems on existence, uniqueness, extension of solutions and continuous dependence of solutions on initial data are presented. The concept of flow is introduced, and then hyperbolicity is considered along with the stable manifold theorem and the Hartman-Grobman theorem. The discussion of hyperbolicity continues in Chapter 10 where many importance for the study of complicated dynamics results is presented: the stable and unstable manifolds theorem for hyperbolic sets, Anosov closing lemma, the shadowing lemma. The chapter concludes with a discussion of structural stability of hyperbolic sets, topological mixing and decomposition, and statistical properties of hyperbolic dynamical systems.

A very important class of quadratic maps is studied in Chapter 11, where attracting periodic orbits, period doubling bifurcations and period-forcing relations, including the Sharkovskiĭ theorem on periodic points of a continuous map of an interval are studied. The Feigenbaum-Misurewicz attractor, hyperbolic Cantor repeller, periodic attractor and Markov repeller are also addressed, and the chapter concludes with a brief discussion of stochastic behavior in quadratic maps. The mechanisms that produce horseshoes and the role played by these in nonlinear dynamics are discussed in Chapter 12. The Birkhoff-Smale theorem and the Poincaré-Melnikov method of detecting homoclinic tangles are also presented. Strange attractors and their properties are briefly addressed in Chapter 13 with a special attention paid to the celebrated Lorenz attractor. Variational methods and twist maps are studied in Chapter 14. The topics include Birkhoff periodic orbits and Aubry-Mather theory for twist maps, invariant circles and regions of instability, periodic points for maps of the cylinder, and existence of closed geodesics on a sphere.

The main goal of the final Chapter 15 is to show how useful the dynamical approach to the problems of uniform distribution and Diophantine approximation of numbers can be. In particular, uniform distribution of squares and polynomials, continued fractions and rational approximation, the Gauss map, quadratic forms in two and three variables are considered.

The first part of the textbook contains numerous useful exercises of different level of difficulty and some problems for further study at more advanced level. Useful suggestions on further and background reading are given, and some helpful facts on metric spaces, differentiability, and Riemann integration in metric spaces are collected in the Appendix. Solutions to many exercises are given in the end of the book, preceded by hints and answers for those readers who do not yield to first difficulties.

The textbook is well written and can be successfully used for teaching at both undergraduate and graduate levels, as well as by the researchers interested in fundamentals of theory of dynamical systems and its recent developments. For the reader who completes work on this book, the natural continuation has to be a more advanced treatise by the same authors [Introduction to the modern theory of dynamical systems, Cambridge Univ. Press, Cambridge (1995; Zbl 0878.58020)], where complete and detailed proofs of many results reported in the book under review can be found. The reader should not be misled by the words “first course” or “introduction” in the titles of the books since both treat dynamical systems with sufficient respect and require significant efforts from the reader who wishes to understand the details and master the methods.

The textbook opens with a nice Introduction whose main purpose is to rouse the interest of the reader to this thrilling and fast developing branch of modern mathematics. From the very beginning, the reader is captivated by skillfully written historical comments and challenged with theoretical exercises and numerical experiments.

The course itself consists of seven chapters. The exposition starts with Chapter 2, where the simplest dynamics exhibited in systems with stable asymptotic behavior is examined. The topics include, but are not limited to, linear maps and linearization, nondecreasing maps of an interval and bifurcations, quadratic maps, fractals. Linear maps and differential equations in the plane and in higher dimension are the subject of the systematic study in Chapter 3. Recurrent behavior of orbits which eventually return close to initial states, but not the exact departure points, is studied in Chapter 4, where rotations of the circle, applications of density and uniform distribution, invertible circle maps, and Cantor phenomena are discussed. The study of recurrence and equidistribution is continued in Chapter 5 for the case of higher dimension. Chapter 6 deals with important aspects of the conservative systems: preservation of the volume and recurrence, Newtonian systems of classical mechanics, and billiards. Simple systems with complicated orbit structure are examined in Chapter 7, where numerous important properties and examples are addressed with a special attention paid to topological transversality, chaos, coding, independence, entropy, and mixing. The discussion of entropy and chaos continues in Chapter 8 which concludes the first part of the book.

The second part of the book opens with Chapter 9 which establishes the link between both parts by emphasizing once again the crucial role played by simple dynamics in understanding more complicated behaviors. The results collected in this chapter are important for the study of dynamical systems: the implicit function theorem, the inverse function theorem, and the smooth contraction principle are among these. Fundamental theorems on existence, uniqueness, extension of solutions and continuous dependence of solutions on initial data are presented. The concept of flow is introduced, and then hyperbolicity is considered along with the stable manifold theorem and the Hartman-Grobman theorem. The discussion of hyperbolicity continues in Chapter 10 where many importance for the study of complicated dynamics results is presented: the stable and unstable manifolds theorem for hyperbolic sets, Anosov closing lemma, the shadowing lemma. The chapter concludes with a discussion of structural stability of hyperbolic sets, topological mixing and decomposition, and statistical properties of hyperbolic dynamical systems.

A very important class of quadratic maps is studied in Chapter 11, where attracting periodic orbits, period doubling bifurcations and period-forcing relations, including the Sharkovskiĭ theorem on periodic points of a continuous map of an interval are studied. The Feigenbaum-Misurewicz attractor, hyperbolic Cantor repeller, periodic attractor and Markov repeller are also addressed, and the chapter concludes with a brief discussion of stochastic behavior in quadratic maps. The mechanisms that produce horseshoes and the role played by these in nonlinear dynamics are discussed in Chapter 12. The Birkhoff-Smale theorem and the Poincaré-Melnikov method of detecting homoclinic tangles are also presented. Strange attractors and their properties are briefly addressed in Chapter 13 with a special attention paid to the celebrated Lorenz attractor. Variational methods and twist maps are studied in Chapter 14. The topics include Birkhoff periodic orbits and Aubry-Mather theory for twist maps, invariant circles and regions of instability, periodic points for maps of the cylinder, and existence of closed geodesics on a sphere.

The main goal of the final Chapter 15 is to show how useful the dynamical approach to the problems of uniform distribution and Diophantine approximation of numbers can be. In particular, uniform distribution of squares and polynomials, continued fractions and rational approximation, the Gauss map, quadratic forms in two and three variables are considered.

The first part of the textbook contains numerous useful exercises of different level of difficulty and some problems for further study at more advanced level. Useful suggestions on further and background reading are given, and some helpful facts on metric spaces, differentiability, and Riemann integration in metric spaces are collected in the Appendix. Solutions to many exercises are given in the end of the book, preceded by hints and answers for those readers who do not yield to first difficulties.

The textbook is well written and can be successfully used for teaching at both undergraduate and graduate levels, as well as by the researchers interested in fundamentals of theory of dynamical systems and its recent developments. For the reader who completes work on this book, the natural continuation has to be a more advanced treatise by the same authors [Introduction to the modern theory of dynamical systems, Cambridge Univ. Press, Cambridge (1995; Zbl 0878.58020)], where complete and detailed proofs of many results reported in the book under review can be found. The reader should not be misled by the words “first course” or “introduction” in the titles of the books since both treat dynamical systems with sufficient respect and require significant efforts from the reader who wishes to understand the details and master the methods.

Reviewer: Yuri V.Rogovchenko (Famagusta)

##### MSC:

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |