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Linear Chaos

  • Textbook
  • © 2011

Overview

Authors:
  1. Karl-G. Grosse-Erdmann

    1. Institut de Mathématique, Université de Mons, Mons, Belgium

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  2. Alfred Peris Manguillot

    1. Institut de Matemàtica Pura i Aplicada, Universitat Politècnica de València, València, Spain

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  • More than 350 exercises and a self-contained exposition makes the book suitable for both classroom use and self-study
  • Presents simplified proofs, as well as improvements, of known results
  • An exhaustive bibliography and the inclusion of a ‘Sources and comments’ section at the end of each chapter provides the reader with ample links to further literature
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (12 chapters)

  1. Front Matter

    Pages I-XI
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  2. Introduction to linear dynamics

    1. Front Matter

      Pages 1-1
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    2. Topological dynamics

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 3-30

    3. Hypercyclic and chaotic operators

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 31-67

    4. The Hypercyclicity Criterion

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 69-88

    5. Classes of hypercyclic and chaotic operators

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 89-135

    6. Necessary conditions for hypercyclicity and chaos

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 137-160

    7. Connectedness arguments in linear dynamics

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 161-178

  3. Selected topics

    1. Front Matter

      Pages 179-179
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    2. Dynamics of semigroups, with applications to differential equations

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 181-211

    3. Existence of hypercyclic operators

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 213-233

    4. Frequently hypercyclic operators

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 235-266

    5. Hypercyclic subspaces

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 267-303

    6. Common hypercyclic vectors

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 305-330

    7. Linear dynamics in topological vector spaces

      • Karl-G. Grosse-Erdmann, Alfred Peris Manguillot
      Pages 331-350

  4. Back Matter

    Pages 351-386
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Keywords

About this book

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.

Reviews

From the reviews:

“The purpose of this well-written book is to provide a complete survey, essentially self-contained, of those results on linear chaos, that is, those results mainly concerned with the study of the behaviour of the iterates of linear transformations … . book under review is nicely written, the material is well motivated and the selected topics are accessible to anyone with a reasonable background in functional analysis at the graduate level. … present monograph constitutes a valuable guide for both students and researchers of the subject.” (E. A. Gallardo-Gutiérrez, Mathematical Reviews, 2013)

“The book under review aims at presenting in an accessible and self-contained way the theory of hypercyclicity and, more generally, the theory of chaos for linear dynamical systems. … It is accessible to advanced undergraduate or beginning graduate students, as well as to nonspecialists who wish to learn about linear dynamical systems. The book covers in great detail what may now be thought of as the basis of the theory, insisting on examples.” (Sophie Grivaux, Zentralblatt MATH, Vol. 1246, 2012)

Authors and Affiliations

  • Institut de Mathématique, Université de Mons, Mons, Belgium

    Karl-G. Grosse-Erdmann

  • Institut de Matemàtica Pura i Aplicada, Universitat Politècnica de València, València, Spain

    Alfred Peris Manguillot

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