Ergodic theory of numbers /

Historically, the field of dynamical systems arises from issues connected with ordinary differential equations. But abstraction, having taken a number of different directions, has birthed a suite of related yet more or less distinct disciplines: topological dynamics, symbolic dynamics, ergodic theory, low-dimensional dynamics, complex dynamics, and chaos theory. Brin and Stuck (both, Univ. of Maryland) manage to introduce concisely all these themes, each already the subject of monographic treatments that run twice the length of the whole volume. And despite the breadth, one finds here major results rigorously treated and substantial applications. By itself, the clean, accessible exposition of the amazing Sharkovsky theorem would justify the acquisition of this book, and one could say the same for the Gauss-Kuzmin-Levy theorem concerning continued fractions and quite a number of others. The authors also explain how Internet search engines like Google use Markov chains and how modified frequency modulation data storage schemes use symbolic dynamics.Although the decimal expansions of numbers like the square root of 2 and pi appear statistically random, no one has ever proved them so. Indeed, despite the familiarity of the decimal expansion process, mathematicians know embarrassingly little. Expanding numbers into decimals has an iterative, dynamical aspect, and ergodic theory studies the intrinsic randomness associated with dynamical iteration. Thus, ergodic theory tells us something of what to expect in the decimal expansions of typical, but not specific, numbers. Dajani (Univ. of Utrecht) and Kraaikamp (Delft Univ. of Technology) tell a somewhat more general story. Besides decimals, other ways exist to expand numbers: continued fractions, Luroth series and generalized Luroth series, beta-series (which generalize decimal expansions). The interaction with ergodic theory carries across this whole spectrum. In a manner that should attract advanced undergraduate mathematics students, this slender book merely sketches the big ideas and sets out some of the simpler proofs, but the recent book by Marius Iosifescu and Kraaikamp, The Metrical Theory of Continued Fractions (2002), offers a very detailed account. Together the books will make a useful pair, with one a stepping stone to the other. ^BSumming Up: Both books--Highly recommended. General readers; lower-division undergraduates through professionals. D. V. Feldman University of New Hampshire