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P-adic Deterministic and Random Dynamics

Andrei Yu. Khrennikov and Marcus Nilsson
Kluwer Academic Publishers
Publication Date: 
Number of Pages: 
Mathematics and Its Applications 574
[Reviewed by
Fernando Q. Gouvêa
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Writing the first book on a new and growing field is a great service to the mathematical community, making the subject much more visible and more accessible to beginners. It is also hard to do well, given that there are no examples to emulate. Hence, we should be grateful to Khrennikov and Nilsson for this book on p-adic dynamics.

The basic idea is to take the theory of discrete complex dynamical systems and move it to the non-archimedean context. Specifically, one looks at iterating certain functions on (the right sort of) p-adic spaces. This is a fairly new field that has generated much interest and lots of theorems. Khrennikov and Nilsson give an overview of this work, very much from their personal point of view and reflecting the work that most interests them. Though far from complete, this is a useful survey.

Mathematicians who have no direct interest in p-adic dynamics might still want to take a look at the book to read the first chapter "On Applications of p-adic Analysis". It's a short survey of various attempts to apply p-adic analysis to physics, biology, and other fields. A lot of these are quite speculative, of course, but they are still interesting.

Research libraries and specialists will want to have this one.

Fernando Q. Gouvêa knows more about the p-adics than he knows about dynamics, so he's glad to have a copy of this book. He is professor of mathematics at Colby College and edits MAA Reviews.

Dedication .- Foreword .- Acknowledgements.

1. On Applications of P-adic Analysis.- 2. P-adic Numbers and P-adic Analysis.- 3. P-adic Dynamical Systems.- 4. Perturbation of Monomial Systems.- 5. Dynamical Systems in Finite Extensions of Qp.- 6. Conjugate Maps.- 7. P-adic Ergodicity.- 8. P-adic Neural Networks.- 9. Dynamics in Ultra-Pseudometric Spaces.- 10. Random Dynamics.- 11. Dynamics of Probability Distributions on the P-adic Mental Space.- 12. Ultrametric Wavelets and Their Applications.- 13. Theory of P-adic Valued Probability.- References.- Index.