Generalized Notions of Continued Fractions Ergodicity and Number Theoretic Applications

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

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; µa; Ta). 7. The Natural Extension of ([0; 1]; ß; µa; 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.

Juan Fernández Sánchez earned his Ph.D. in mathematics from the University of Almería (Spain) in 2010. His research interests are in dependence modeling and copulas, dynamical systems, singular functions, and number theory. Jerónimo López-Salazar Codes completed his doctoral work under the supervision of Professors José María Martínez Ansemil and Socorro Ponte at Universidad Complutense de Madrid (Spain) and obtained his Ph.D. degree in 2013. He currently works at Universidad Politécnica de Madrid (Spain). His research is mainly devoted to infinite dimensional holomorphy and lineability. Juan B. Seoane Sepúlveda earned his first Ph.D. at the Universidad de Cádiz (Spain) jointly with Universität 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.