Generalized Notions of Continued Fractions
Ergodicity and Number Theoretic Applications
1st Edition
Published on 20. July 2023
Book
Hardback
142 pages
978-1-032-51678-3 (ISBN)
Description
Ancient times witnessed the origins of the theory of continued fractions. Throughout time, mathematical geniuses such as Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, or Euler have made significant contributions to the development of this famous theory, and it continues to evolve today, especially as a means of linking different areas of mathematics.
This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed.
Features
Suitable for graduate students and senior researchers
Written by international senior experts in number theory
Contains the basic background, including some elementary results, that the reader may need to know before hand, making it a self-contained volume
This book, whose primary audience is graduate students and senior researchers, is motivated by the fascinating interrelations between ergodic theory and number theory (as established since the 1950s). It examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. After deriving invariant ergodic measures for each of the underlying transformations on [0,1] it is shown that any of the famous formulas, going back to Khintchine and Levy, carry over to more general settings. Complementing these results, the entropy of the transformations is calculated and the natural extensions of the dynamical systems to [0,1]2 are analyzed.
Features
Suitable for graduate students and senior researchers
Written by international senior experts in number theory
Contains the basic background, including some elementary results, that the reader may need to know before hand, making it a self-contained volume
More details
Series
Language
English
Place of publication
United Kingdom
Publishing group
Taylor & Francis Ltd
Target group
College/higher education
Postgraduate
Illustrations
7 s/w Abbildungen, 2 farbige Abbildungen, 7 s/w Zeichnungen, 2 farbige Zeichnungen
2 Line drawings, color; 7 Line drawings, black and white; 2 Illustrations, color; 7 Illustrations, black and white
Dimensions
Height: 240 mm
Width: 161 mm
Thickness: 13 mm
Weight
402 gr
ISBN-13
978-1-032-51678-3 (9781032516783)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Juan Fernandez Sanchez | Jeronimo Lopez-Salazar Codes | Juan B. Seoane Sepulveda
Generalized Notions of Continued Fractions
Ergodicity and Number Theoretic Applications
E-Book
07/2023
1st Edition
Chapman & Hall/CRC
€231.99
Available for download
Juan Fernandez Sanchez | Jeronimo Lopez-Salazar Codes | Juan B. Seoane Sepulveda
Generalized Notions of Continued Fractions
Ergodicity and Number Theoretic Applications
E-Book
07/2023
1st Edition
Chapman & Hall/CRC
€231.99
Available for download
Persons
Juan Fernandez Sanchez earned his Ph.D. in mathematics from the University of Almeria (Spain) in 2010. His research interests are in dependence modeling and copulas, dynamical systems, singular functions, and number theory.
Jeronimo Lopez-Salazar Codes completed his doctoral work under the supervision of Professors Jose Maria Martinez Ansemil and Socorro Ponte at Universidad Complutense de Madrid (Spain) and obtained his Ph.D. degree in 2013. He currently works at Universidad Politecnica de Madrid (Spain). His research is mainly devoted to infinite dimensional holomorphy and lineability.
Juan B. Seoane Sepulveda earned his first Ph.D. at the Universidad de Cadiz (Spain) jointly with Universitaet Karlsruhe (Germany) in 2005. His received his second Ph.D. at Kent State University (Kent, Ohio, USA) in 2006. His main interests include Real and Complex Analysis, Operator Theory, Number Theory, Mathematical Modeling, Mathematical Biology, Geometry of Banach spaces, History of Mathematics, and Lineability. He is the author of over 200 scientific publications, including several books. He is currently a professor at Universidad Complutense de Madrid, where he also holds the position of director of the Master's in Advanced Mathematics.
Wolfgang Trutschnig obtained his Ph.D. at the Vienna University of Technology, Austria, in 2006. He is currently the professor for stochastics and director of the IDA Lab at the Paris Lodron University Salzburg (PLUS) and mainly works in dependence modeling and nonparametric statistics with regular excursions to dynamical systems, fractals and ergodic theory.
Jeronimo Lopez-Salazar Codes completed his doctoral work under the supervision of Professors Jose Maria Martinez Ansemil and Socorro Ponte at Universidad Complutense de Madrid (Spain) and obtained his Ph.D. degree in 2013. He currently works at Universidad Politecnica de Madrid (Spain). His research is mainly devoted to infinite dimensional holomorphy and lineability.
Juan B. Seoane Sepulveda earned his first Ph.D. at the Universidad de Cadiz (Spain) jointly with Universitaet Karlsruhe (Germany) in 2005. His received his second Ph.D. at Kent State University (Kent, Ohio, USA) in 2006. His main interests include Real and Complex Analysis, Operator Theory, Number Theory, Mathematical Modeling, Mathematical Biology, Geometry of Banach spaces, History of Mathematics, and Lineability. He is the author of over 200 scientific publications, including several books. He is currently a professor at Universidad Complutense de Madrid, where he also holds the position of director of the Master's in Advanced Mathematics.
Wolfgang Trutschnig obtained his Ph.D. at the Vienna University of Technology, Austria, in 2006. He is currently the professor for stochastics and director of the IDA Lab at the Paris Lodron University Salzburg (PLUS) and mainly works in dependence modeling and nonparametric statistics with regular excursions to dynamical systems, fractals and ergodic theory.
Content
1. Generalized Lehner Continued Fractions. 2. a-modified Farey Series. 3. Ergodic Aspects of the Generalized Lehner Continued Fractions. 4. The a-simple Continued Fraction. 5. The Generalized Khintchine Constant. 6. The Entropy of the System ([0; 1]; ss; ?a; Ta). 7. The Natural Extension of ([0; 1]; ss; ?a; Ta). 8. The Dynamical System ([0; 1]; ss; va;Qa). 9. Generalized Hirzebruch-Jung Continued Fractions. 10. The Entropy of ([0; 1]; ss; ?a;Ha). 11. The Natural Extension of ([0; 1]; ss; ?a;Ha). 12. A New Generalization of the Farey Series.