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Applied algebraic dynamics. (English) Zbl 1184.37002
de Gruyter Expositions in Mathematics 49. Berlin: Walter de Gruyter (ISBN 978-3-11-020300-4/pbk). xxiv, 533 p. (2009).
Barely two years after the publication of Silverman’s influential book ‘The arithmetic of dynamical systems’; the release of this new volume reflects the vitality and breadth of the research at the interface of dynamical systems and number theory.
The book ‘Applied algebraic dynamics’ occupies a place of its own; with hardly any overlap with existing literature. It deals with dynamical systems over sets endowed with an algebraic structure; providing a highly original blend of theory and applications. These reflect closely the authors’ research interests (their names appear in a quarter of the four hundred or so items in the bibliography). The theoretical material is centred on non-archimedean dynamical systems; with emphasis on their analytic and ergodic-theoretic properties. The rest of the book is concerned with applications outside mathematics: computer sciences; quantum theory; cognitive sciences; and genetics. The common feature of such a diverse (and surprising) list; is the applicability of non-Archimedean methods to approach the problems at hand.
The book is aimed at a heterogeneous readership; and the exposition is adjusted accordingly. While the theoretical sections are written for mathematicians; the application sections adopt a more informal style; and are; to some degree; self-contained. However; apart from a few chapters near the end; one is never too far away from formal mathematical statements; and a fair deal of mathematical maturity will be required throughout.
The book begins with a succinct review of essential constructs of algebra and number theory. Dynamical systems first appear in the second chapter; mostly defined on finite commutative groups and rings. Requiring the existence of an ergodic polynomial map restricts considerably the nature of the ring. The necessary \(p\)-adic analysis is developed in the following chapter; as a prelude to \(p\)-adic ergodic theory. Some concepts relevant to applications (modular differentiability; compatibility; a classification of locally analytic functions) are introduced.
The \(p\)-adic ergodic theory is first developed for one-dimensional monomial dynamical maps; where conditions for unique ergodicity on spheres are derived. In higher dimensions; various results on measure preservation and ergodicity are derived for 1-Lipschitz maps; whose modular properties are studied thoroughly; in view of applications. Monomial dynamical systems reappear in chapter 5; where the structure of their cycles and fuzzy cycles (cyclic arrangements of spheres) is studied.
Chapter 6 deals with ergodic polynomials over finite non-commutative groups with operators. Again the existence of an ergodic polynomial imposes strong constraints on the group; and various classification results are obtained.
The applications of \(p\)-adic ergodic theory begin with automata theory; and quickly turns to digital computers. The main theme is that some basic CPU instructions; both numerical and logical; may be represented as functions which are uniformly continuous with respect to the 2-adic metric. As an application to combinatorics; uniformly differentiable \(p\)-adic maps are used to construct Latin squares. Substantial applications of ergodic \(p\)-adic maps are found in the construction and analysis of pseudo-random numbers and stream ciphers. The geometrical device of representing sequences as sets of points in the unit square provides a useful complement to the mathematics.
A \(p\)-adic probability theory is presented in Chapter 12; which is needed for the development of a non-Archimedean version of quantum mechanics. The authors first analyse foundational issues; and then provide the basic constructs; guiding the reader thorough the peculiarities of this theory (e.g.; the existence of negative probabilities). The chapter on \(p\)-adic valued quantisation gives a very concise account of a rather large body of research; which will require access to the relevant literature.
The last four chapters have a rather different flavour; and seems more tentative. Various quantities relevant to applications (such as the “mental spaces” in cognitive sciences) have; in some cases at least; a natural hierarchical structure. Such a structure suggests representations on trees. In this context; the choice of specific non-Archimedean rings seems to be dictated more by mathematical convenience then by experimental evidence. However; the authors’ main concern here is not to obtain specific results; but rather to promote the use of ultrametric ideas.

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37Pxx Arithmetic and non-Archimedean dynamical systems
11S82 Non-Archimedean dynamical systems
60B99 Probability theory on algebraic and topological structures
81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
92D10 Genetics and epigenetics