Spaces of dynamical systems
\textit{S. Yu. Pilyugin}'s ``Spaces of Dynamical Systems'' provides a formal introduction to the main objects and results of the modern theory of dynamical systems, concentrating on hyperbolicity and structural stability. This text is intended to complement the author's earlier text ``Introduction to structurally stable systems of differential equations'' [Translated from the Russian by the author. Basel etc: Birkhäuser Verlag. (1992; Zbl 0747.34031)]. These two books have perceptible overlap in content, though neither text requires the other as a prerequisite. The chief dissimilarity between the two texts is that the earlier text focused on the dynamics of smooth flows, while much of the current work emphasizes discrete time dynamical systems, only presenting the results for flows which are analogues of the discrete time case. In addition to classical statements and examples, the book introduces the reader to a few specialized topics, including shadowing of pseudotrajectories. The style is rigorous theorem/proof mathematical format, with a detailed and clear exposition. The main strength of this book is that the author includes complete proofs of results that other publications frequently leave to the reader. The book foregoes lengthy introductions in favor of contextualizing results by concise historical comments. NEWLINENEWLINENEWLINE``Spaces of Dynamical Systems'' assumes a solid command of topics from an undergraduate education in math: functional analysis, topology, linear algebra, calculus, differential equations, and analysis. This writing would be best suited for early graduate students or professional mathematicians new to the field of dynamics. There are a few explicit exercises embedded in the text, however, supplementation might be advantageous if the book is to be used in a course. This book contains twelve chapters and two appendices (together a total of 221 pages), and 7 grayscale figures.NEWLINENEWLINEThe first three chapters set the groundwork for the remainder of the text. The first chapter, ``Dynamical Systems'', establishes the basic objects studied in both discrete and continuous dynamical systems including trajectories and orbits, periodic and fixed points, and invariant sets. The author uses the standard example, the shift map on the space of binary sequences, to exemplify these concepts and to foreshadow the topics in Chapter 9 on the Smale horseshoe and chaotic sets. Embeddings of discrete dynamical systems into flows are introduced in consideration of the subsequent topic: local Poincaré diffeomorphisms, and the meaningful case of flows with periodic trajectories is highlighted. The second chapter defines topologies on the spaces of homeomorphisms and diffeomorphisms via \(C^0\) and \(C^1\) metrics respectively, and topologies on spaces of autonomous systems of ordinary differential equations are considered. Finally, the reader is introduced to generic properties by means of residual sets. Chapter 3 covers the established notions of equivalence for dynamical systems: topological conjugacy for maps and flows and topological equivalence of flows, allowing for the formalization of the notion of structural stability.NEWLINENEWLINEChapters 4--8 address hyperbolicity as one of the fundamental notions in the theory of structural stability. The reader is taken through the fundamental ideas surrounding hyperbolicity, in particular the Grobman-Hartman linearization theorem and the stable manifold theorem. The reader is introduced to these topics in increasing levels of generality, first starting with hyperbolic fixed points, then studying hyperbolic periodic points, and finally looking at general hyperbolic sets. Transversality, an important idea in dynamics, is treated in detail. Lastly, several important classes of dynamical systems are studied including Kupka-Smale diffeomorphisms, Smale's Axiom A diffeomorphisms, and Anosov diffeomorphisms.NEWLINENEWLINEChapters 9--12 examine additional essential topics in the study of dynamics. The first section of Chapter 9, ``Smale's horseshoe and chaos'', describes Smale's horseshoe and proves that it conjugates with the shift map introduced in Chapter 1. The second section is on chaotic sets, where the invariant sets of the shift map and the horseshoe map are revealed to be chaotic. Chapter 10 deals with results related to the ``Closing Lemma'', which essentially states that a positive orbit that comes back near to the initial point can be closed to a periodic orbit under small perturbation in the \(C^1\)-topology. This theorem is stated without proof, instead referring to the standard references. Several related results are listed, one of which is used in the proof of Mañé's theorem of structural stability, outlined in Appendix A. Chapter 11, on \(C^0\)-generic properties of dynamical systems, encompasses properties of dynamical systems that hold over residual subsets of spaces of these systems. Essential topics such as tolerance stability, attractors, Lyapunov stability, and stable attractors are treated. Pilyugin spends Chapter 12 on ``Shadowing of pseudotrajectories'', a vital topic for those wishing to conduct numerical investigations of dynamical systems. Pilyugin defines the shadowing property and the inverse shadowing property, and he spends the rest of the chapter developing detailed proofs of associated theorems.NEWLINENEWLINEThe author includes in this text two appendices. The first one, Appendix A, offers an outline of the proof of the Mañé theorem that a structurally stable diffeomorphism satisfies Axiom A. Appendix B is a collection of lectures on the history of differential equations and dynamical systems. The lectures start with Newton and his influence on the field, continue through the nonlocal qualitative theory of dynamical systems, structural stability, and chaotic behavior, contextualizing the main ideas in this text.NEWLINENEWLINEAltogether, this text is for serious readers that would like a clean and rigorous introduction to the results and methods of proof in dynamical systems. It would make a valuable addition as a reference text to the collection of any academician working in related fields.
scientific article; zbMATH DE number 6023866