## Linear ChaosIt 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. |

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### Contents

Part II Selected topics | 179 |

Appendix A Prerequisites | 351 |

Appendix B Spectral theory | 361 |

367 | |

379 | |

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### Common terms and phrases

0-neighbourhood adjoint assertions are equivalent Banach space Bayart Birkhoff chaotic operators Chapter Co-semigroup common hypercyclic vectors compact operator complex composition operator condition Conejero consider converges Corollary countable deduce dense orbit dense range dense set dense subset dynamical system eigenvalue eigenvector entire function Example Exercise exists finite following assertions Grivaux Grosse-Erdmann Hahn–Banach theorem hence Hilbert space Hint holomorphic function hypercyclic subspace Hypercyclicity Criterion implies infinite-dimensional separable Lemma linear dynamics Math metric space mixing operator Moreover nk)k of positive nonempty open sets nonempty open subsets nonzero obtain open sets operator norm periodic points Peris polynomial positive integers proof of Theorem Proposition result satisfies the Hypercyclicity Section semigroup seminorms separable Banach space separable Fréchet space sequence space Show span Suppose Tn)n topological vector space topologically transitive weak mixing weakly mixing weighted shift