Multidimensional Continued Fractions. By Fritz Schweiger. Oxford Science Publications

The field of multidimensional continued fraction algorithms (MCFAs) is at the magnificent crossroads of number theory and ergodic theory. The number theoretical aspects covered in this book are convergence, periodicity and Diophantine approximations. Moreover, it covers their dynamical aspects such as ergodic and metric properties. The field is a rich source of examples in ergodic theory and many dynamical systems have underlying MCFAs. There has been much progress made on the subject in the last few decades and it deserves texts such as the present one. So far the books dedicated to MCFAs are a survey by Brentjes [B], which concerns their arithmetical and computational aspects, and an earlier book by Schweiger [S], which regards their ergodic and metric properties. There is an intersection between Schweiger’s new book and his earlier one. The regular continued fraction expresses an irrational number s as