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Reversibility in Dynamics and Group Theory

Book Description

Reversibility is a thread woven through many branches of mathematics. It arises in dynamics, in systems that admit a time-reversal symmetry, and in group theory where the reversible group elements are those that are conjugate to their inverses. However, the lack of a lingua franca for discussing reversibility means that researchers who encounter the concept may be unaware of related work in other fields. This text is the first to make reversibility the focus of attention. The authors fix standard notation and terminology, establish the basic common principles, and illustrate the impact of reversibility in such diverse areas as group theory, differential and analytic geometry, number theory, complex analysis and approximation theory. As well as showing connections between different fields, the authors' viewpoint reveals many open questions, making this book ideal for graduate students and researchers. The exposition is accessible to readers at the advanced undergraduate level and above.

Table of Contents

  1. Coverpage
  2. Series information
  3. Title page
  4. Copyright page
  5. Contents
  6. Preface
  7. 1 Origins
    1. 1.1 Origins in dynamical systems
    2. 1.2 Origins in finite group theory
    3. 1.3 Origins in hidden dynamics
    4. 1.4 The reversibility problem
  8. 2 Basic ideas
    1. 2.1 Reversibility
    2. 2.2 Reformulations
    3. 2.3 Signed groups
    4. 2.4 Whither next?
  9. 3 Finite groups
    1. 3.1 Reversers of finite order
    2. 3.2 Dihedral groups
    3. 3.3 Symmetric groups
    4. 3.4 Alternating groups
    5. 3.5 Group characters
    6. 3.6 Characters and reversible elements
    7. 3.7 Characters and strongly-reversible elements
    8. 3.8 Examples
    9. 3.9 Free groups
  10. 4 The classical groups
    1. 4.1 The classical groups
    2. 4.2 The general linear group
    3. 4.3 The orthogonal group
    4. 4.4 The unitary group
    5. 4.5 Summary
  11. 5 Compact groups
    1. 5.1 Reversibility in compact groups
    2. 5.2 Compact Lie groups
    3. 5.3 The special unitary group
    4. 5.4 Compact symplectic groups
    5. 5.5 The spinor groups
  12. 6 Isometry groups
    1. 6.1 Isometries of spherical, Euclidean, and hyperbolic space
    2. 6.2 Hyperbolic geometry in two and three dimensions
    3. 6.3 Euclidean isometries
    4. 6.4 Hyperbolic isometries
  13. 7 Groups of integer matrices
    1. 7.1 Conjugacy to rational canonical form
    2. 7.2 Integral quadratic forms
    3. 7.3 Conway’s topograph
    4. 7.4 Gauss’ method for definite forms
    5. 7.5 Elliptic elements of GL(2, Z)
    6. 7.6 Centralisers
    7. 7.7 Reversible elements
    8. 7.8 Second associated form
    9. 7.9 Gauss’ method for indefinite forms
    10. 7.10 Another way
    11. 7.11 Cyclically-reduced words
  14. 8 Real homeomorphisms
    1. 8.1 Involutions
    2. 8.2 Conjugacy
    3. 8.3 Reflectional and rotational symmetries
    4. 8.4 Reversible elements
    5. 8.5 Products of involutions and reversible elements
  15. 9 Circle homeomorphisms
    1. 9.1 Involutions
    2. 9.2 Conjugacy
    3. 9.3 Reversible elements
    4. 9.4 Strongly-reversible elements
    5. 9.5 Products of involutions
  16. 10 Formal power series
    1. 10.1 Power series structures
    2. 10.2 Elements of finite order
    3. 10.3 Conjugacy
    4. 10.4 Centralisers
    5. 10.5 Products of involutions modulo X[sup(4)]
    6. 10.6 Strongly-reversible elements
    7. 10.7 Products of involutions
    8. 10.8 Reversible series
    9. 10.9 Reversers
    10. 10.10 Products of reversible series
  17. 11 Real diffeomorphisms
    1. 11.1 Involutions
    2. 11.2 Fixed-point-free maps
    3. 11.3 Centralisers
    4. 11.4 Reversibility of order-preserving diffeomorphisms
    5. 11.5 Reversibility in the full diffeomorphism group
    6. 11.6 Products of involutions and reversible elements
  18. 12 Biholomorphic germs
    1. 12.1 Elements of finite order
    2. 12.2 Conjugacy
    3. 12.3 Écalle–Voronin theory
    4. 12.4 Roots
    5. 12.5 Centralisers and flows
    6. 12.6 Reversible and strongly-reversible elements
    7. 12.7 The order of a reverser
    8. 12.8 Examples
  19. References
  20. List of frequently used symbols
  21. Index of names
  22. Subject index