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Ergodic theory of numbers. (English) Zbl 1033.11040
The Carus Mathematical Monographs 29. Washington, DC: The Mathematical Association of America (MAA) (ISBN 0-88385-034-6/hbk). x, 190 p. \$ 39.95 (2002).
In this book two fields of Mathematics, Number theory and Ergodic theory, are interacted. Namely, the number-theoretic questions are answered with the help of Ergodic theory. In Chapter 1 the decimal expansion is characterized and the concept of probability measure is introduced. Another way to represent numbers, the continued fraction algorithm, is also mentioned. In Chapter 2, the decimal expansions are generalized to introduce $n$-ary expansions, Lüroth series, generalized Lüroth series (GLS) and beta-expansions. In Chapter 3 some of the basic results in ergodic theory are presented. The existence of a T-invariant measure allows us to use ergodic theory to answer many number-theoretical questions. In Chapter 4 a deep relationship between GLS expansion and beta-expansion is shown. In Chapter 5 many metric and Diophantine properties of the regular and many other continued fraction expansions are mentioned. In Chapter 6 the concept of entropy is introduced. How to calculate the entropy, the Shannon-McMillan-Breiman theorem and Saleski’s theorem are mentioned.
##### MSC:
 11K50 Metric theory of continued fractions 11-02 Research monographs (number theory) 11K55 Metric theory of other number-theoretic algorithms and expansions 37A45 Relations of ergodic theory with number theory and harmonic analysis