Ergodic theory of fibred systems and metric number theory. (English) Zbl 0819.11027

Schweiger, Fritz

Oxford: Clarendon Press. xiii, 295 p. (1995).

The present monograph develops the idea of the application of ergodic theory to number expansions. As is well known there are many algorithms, which eventually may be periodic, useful to write a real number \(x\) as an expansion \([a_ 1, a_ 2, \dots]\). Usually this expansion is obtained by iteration of a map \(T\). For instance the elementary continued fraction expansion is given by the iteration of the map \(Tx= (1/x)- [1/x]\). Then the ergodic theory is useful in the study of number-expansion-algorithms governed by the iteration of suitable maps \(T\) and the corresponding problems to be solved are as follows: Is \(T\) ergodic and/or conservative? Does \(T\) admit an invariant measure absolutely continuous with respect to Lebesgue measure?
The book begins with the notion of fibred system, and its developments are divided into 31 chapters (some titles are as follows: Continued fractions, Multidimensional continued fractions, Invariant measures, Kuzmin equations, Dual algorithms, Diophantine problems, The quadratic map, and others).
The author presents an updated and extended bibliography, selected firstly, to quote the sources from which the material covered in the book is taken, and secondly, to propose further references to the reader.
The book is an excellent piece of work and scholarship.

Reviewer: S.L.Kalpazidou (Thessaloniki)

MSC:

11K55	Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11-02	Research exposition (monographs, survey articles) pertaining to number theory
28D20	Entropy and other invariants
11K50	Metric theory of continued fractions
28D05	Measure-preserving transformations
28-02	Research exposition (monographs, survey articles) pertaining to measure and integration
37A99	Ergodic theory
37B99	Topological dynamics

Keywords:

continued fractions; multidimensional continued fraction; invariant measures; dual algorithms; diophantine problems; Kuzmin equations; ergodic theory; number expansions; iteration; Lebesgue measure; fibred system; quadratic map; bibliography