Multidimensional Continued Fractions

by Fritz Schweiger

Overview

This work offers an overview of various aspects of multidimensional continued fractions, which are here defined through iteration of piecewise fractional linear maps. This includes the algorithms of Jacobi-Perron, Guting, Brun, and Selmer but it also includes continued fractions on simplices which are related to interval exchange maps or the Parry-Daniels map. New classes of subtractive algorithms are also included and the metric properties of these algorithms can be therefore investigated by methods of ergodic theory. The recent connection between multiplicative ergodic theory and Diophantine approximation presented, as well as several results on convergence and Perron's approach to periodicity, which has never appeared in print despite being published in 1907. Further chapters include the basic properties of continued fractions in the complex plane, connections with Hausdorff dimension and the Kuzmin theory for multidimensional maps.

Product Details

ISBN-13:	9780198506867
Publisher:	Oxford University Press
Publication date:	10/13/2000
Pages:	244
Product dimensions:	6.10(w) x 9.10(h) x 0.70(d)

Table of Contents

1. Piecewise fractional linear maps2. Brentjes' formalism3. Ergodic theory of fibred systems4. Jacobi-Perron algorithm4.1. Basic definitions4.2. Convergence results4.3. Ergodic properties5. Güting's algorithm6.1. Subtractive version6.2. Multiplicative acceleration7. Selmer's algorithm7.1. Subtractive version7.2. The division algorithm8. Generalized substractive algorithms9. Fully subtractive algorithms10. Skew products11. Continued fractions on simplices11.1. Fractional linear maps11.2. Interval exchange transformations12. The convergence method of Greiter13. Periodic expansions14. Convergence and rational dependence15. Diophantine approximation15.1. Exponents of approximation15.2. Perron's identity15.3. Results on convergence16. The second Lyapunov exponent17. Periodic Jacobi-Perron algorithms18. Convergence for periodic expansions19. Volume as a measure of approximation20. Complex continued fractions21. The Poincaré alogrithm22. Dimension of exceptional sets22.1. Introduction22.2. The Dirichlet series of a set23. A Kuzmin theorem24. Some applicationsReferencesIndex

Editorial Reviews

"It is thanks to F. Schweiger that the extensive literature on multidimensional continued fractions, especially those which can be described by fractional linear maps can now be seen from a more systematic point of view and it is thanks to the publishing company for having won a real expert for this project. . .The book is very carefully written and indispensable for all experts working in this field."—onatshefte Fuer Mathematik

From the Publisher

Mathematician Fritz Schweiger, whose academic affiliation is not provided, provides an introduction to a field of research that has seen remarkable progress in recent decades, concentrating on multidimensional continued fractions which can be described by fractional linear maps or equivalently by a set of ( + 1) x ( + 1) matrices. Addressing the question of periodicity, he refines the problem of convergence to the question of whether these algorithms give "good" simultaneous Diophantine approximations. He notes that these algorithms are not likely to provide such "good" approximations which satisfy the -dimensional Dirichlet property. Also studied are the ergodic properties of these maps. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews