by R S MacKay (University of Warwick, UK)

This book is adapted and revised from the author's seminal PhD thesis, in which two forms of asymptotically universal structure were presented and explained for area-preserving maps. Area-preserving maps are the discrete-time analogue of two degree-of-freedom Hamiltonian systems. How they work and much of their dynamics are described in this book. The asymptotically universal structure is found on small scales in phase-space and long time-scales. The key to understanding it is renormalisation, that is, looking at a system on successively smaller phase-space and longer time scales. Having presented this idea, the author briefly surveys the use of the idea of renormalisation in physics. The renormalisation picture is then presented as the key to understanding the transition from regular to chaotic motion in area-preserving maps. Although written ten years ago, the subject matter continues to interest many today. This updated version will be useful to both researchers and students.

• Introduction to Area Preserving Maps:
  • Conservative Systems and Area Preserving Maps
  • Periodic Orbits
  • Invariant Circles
  • Stochastic Behaviour
• Introduction to Renormalisation:
  • Renormalisation Physics
  • Renormalisation in Dynamical Systems
  • Renormalisation Techniques
• Period Doubling in Area Preserving Maps:
  • Period Doubling Sequences
  • Renormalisation
  • The Universal One Parameter Family
  • Appendices
• Renormalisation for Invariant Circles:
  • Renormalisation
  • Fixed Point Analysis
  • Simple Fixed Point
  • Critical Fixed Point
  • The Universal One Parameter Family
  • Discussion
  • Renormalisation for Maps on a Circle