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Über dieses Buch

It is commonly believed that chaos is linked to non-linearity, however many (even quite natural) linear dynamical systems exhibit chaotic behavior. The study of these systems is a young and remarkably active field of research, which has seen many landmark results over the past two decades. Linear dynamics lies at the crossroads of several areas of mathematics including operator theory, complex analysis, ergodic theory and partial differential equations. At the same time its basic ideas can be easily understood by a wide audience. Written by two renowned specialists, Linear Chaos provides a welcome introduction to this theory. Split into two parts, part I presents a self-contained introduction to the dynamics of linear operators, while part II covers selected, largely independent topics from linear dynamics. More than 350 exercises and many illustrations are included, and each chapter contains a further ‘Sources and Comments’ section. The only prerequisites are a familiarity with metric spaces, the basic theory of Hilbert and Banach spaces and fundamentals of complex analysis. More advanced tools, only needed occasionally, are provided in two appendices. A self-contained exposition, this book will be suitable for self-study and will appeal to advanced undergraduate or beginning graduate students. It will also be of use to researchers in other areas of mathematics such as partial differential equations, dynamical systems and ergodic theory.

Inhaltsverzeichnis

Frontmatter

Introduction to linear dynamics

Frontmatter

Chapter 1. Topological dynamics

Abstract
This chapter introduces the reader to the fundamental concepts of the theory of (not necessarily linear) dynamical systems. Fundamental concepts such as topologically transitive, chaotic, weakly mixing and mixing maps are defined and illustrated with typical examples. The Birkhoff transitivity theorem is derived as a crucial tool for showing that a map has a dense orbit. Moreover we obtain several characterizations of weakly mixing maps that will be of great significance later on.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 2. Hypercyclic and chaotic operators

Abstract
In this chapter, the notions and results from the first chapter are revisited in the context of linearity. We introduce the notion of a hypercyclic operator and that of a chaotic operator. Among other things it is proved that the classical operators of Birkhoff, MacLane and Rolewicz are chaotic; it is shown that every hypercyclic operator possesses a dense subspace all of whose nonzero vectors are hypercyclic (the Herrero–Bourdon theorem), and that linear dynamics can be as complicated as nonlinear dynamics. We begin the chapter with an introduction to Fréchet spaces since they provide the setting for some important chaotic operators.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 3. The Hypercyclicity Criterion

Abstract
This chapter presents several criteria for hypercyclicity, weak mixing, mixing and chaos, in increasing order of sophistication. It culminates in the Hypercyclicity Criterion, which is discussed in detail. We prove, among other things, that the Hypercyclicity Criterion characterizes the weak mixing property.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 4. Classes of hypercyclic and chaotic operators

Abstract
In this chapter, some important classes of hypercyclic and chaotic operators are described: weighted (unilateral and bilateral) shift operators on sequence spaces, differential operators and composition operators on the space of all holomorphic functions on a domain, and adjoint multiplication operators and composition operators on the Hardy space.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 5. Necessary conditions for hypercyclicity and chaos

Abstract
In this chapter we discuss the spectral properties of hypercyclic and chaotic operators. We obtain, in particular, Kitai’s theorem that each connected component of the spectrum of a hypercyclic operator meets the unit circle. As an application we derive properties that preclude hypercyclicity or chaos, and we obtain classes of operators that do not contain any hypercyclic operator.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 6. Connectedness arguments in linear dynamics

Abstract
This chapter presents some of the deepest, most beautiful and most useful results from linear dynamics. We obtain Ansari’s theorem that every power of a hypercyclic operator is hypercyclic, the Bourdon–Feldman theorem that every somewhere dense orbit is (everywhere) dense, the Costakis–Peris theorem that every multi-hypercyclic operator is hypercyclic, the León–Müller theorem that any unimodular multiple of a hypercyclic operator is hypercyclic, and the Conejero–Müller–Peris theorem that every operator in a hypercyclic semigroup is hypercyclic.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Selected topics

Frontmatter

Chapter 7. Dynamics of semigroups, with applications to differential equations

Abstract
This chapter discusses the continuous analogue of hypercyclic and chaotic operators in the form of semigroups. While the theories run parallel in great parts, hypercyclic and chaotic semigroups have important applications to partial differential equations and to infinite linear systems of ordinary differential equations. Representative examples are discussed.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 8. Existence of hypercyclic operators

Abstract
In this chapter we obtain, among other things, the Ansari–Bernal theorem that every infinite-dimensional separable Banach space supports a hypercyclic operator. In contrast, some infinite-dimensional separable Banach spaces do not support any chaotic operator. We also discuss here the richness of the set of hypercyclic operators in two ways: it forms a dense set in the space of all operators when endowed with the strong operator topology; and it is shown that any linearly independent sequence of vectors appears as the orbit under a hypercyclic operator.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 9. Frequently hypercyclic operators

Abstract
The contents of this chapter are motivated by recent work on the application of ergodic theory to linear dynamics. While the technical difficulties involved prevent us from studying these tools here, we will discuss a new concept that has come out of these investigations, the frequently hypercyclic operators. We obtain, among other things, a Frequent Hypercyclicity Criterion, we show that the three classical hypercyclic operators are frequently hypercyclic, and we revisit properties of hypercyclic and chaotic operators in the context of frequent hypercyclicity.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 10. Hypercyclic subspaces

Abstract
This chapter is devoted to the question of whether there is, for a given operator, an infinite-dimensional closed subspace all of whose nonzero vectors are hypercyclic. Such a subspace is called a hypercyclic subspace. We give two proofs of Montes’ theorem that provides a sufficient condition for the existence of hypercyclic subspaces. The first proof provides an explicit construction via basic sequences, the second one relies on the study of left-multiplication operators. We also obtain conditions that prevent the existence of hypercyclic subspaces; as an application we show that Rolewicz’s operators do not have hypercyclic subspaces.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 11. Common hypercyclic vectors

Abstract
This chapter studies the existence of common hypercyclic vectors for families of operators. While countable families of hypercyclic operators on the same space automatically possess common hypercyclic vectors, this is no longer the case for uncountable families. The Common Hypercyclicity Criterion provides a sufficient condition for a (one-parameter) family of operators to admit a common hypercyclic vector. We study, in particular, common hypercyclic vectors for multiples of a given operator. We close the chapter with a study of common hypercyclic subspaces.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Chapter 12. Linear dynamics in topological vector spaces

Abstract
This chapter treats hypercyclicity and linear chaos in their most natural (and most general) setting, that of topological vector spaces. After a brief introduction to such spaces we revisit many of the results previously obtained in the book and show that they hold in great generality. We also derive dynamical transference principles which allow us to transfer the dynamical properties of operators on F-spaces to operators on general topological vector spaces.
Karl-G. Grosse-Erdmann, Alfred Peris Manguillot

Backmatter

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