Multidimensional Continued Fractions
“Multidimensional Continued Fractions” by Fritz Schweiger, published by Oxford University Press in 2000, offers a comprehensive overview of multidimensional continued fractions, defined through the iteration of piecewise fractional linear maps. This edition, comprising 234 pages, delves into various algorithms, including those of Jacobi-Perron, Güting, Brun, and Selmer, while also exploring continued fractions on simplices related to interval exchange maps and the Parry-Daniels map.
Readers will find an in-depth examination of new classes of subtractive algorithms and their metric properties, investigated through ergodic theory methods. The book discusses the recent connections between multiplicative ergodic theory and Diophantine approximation, alongside results on convergence and Perron’s approach to periodicity. Additional chapters cover the fundamental properties of continued fractions in the complex plane, as well as their connections to Hausdorff dimension and Kuzmin theory for multidimensional maps.
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The book gives an up to date 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, Güting, 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 book 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.
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