Generalized Notions of Continued Fractions : Ergodicity and Number Theoretic Applications

There is no clear sense of when the continued fraction was originally conceived of. It is likely that one of the first authors who, indirectly, suggested this notion was Euclid (c. 300 BC) via his famous algorithm (the oldest nontrivial algorithm that has survived to the present day) in the seventh book of his Elements. Since then, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, and Euler have developed this 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. 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 Cover Half Title Series Page Title Page Copyright Page Contents Preface Authors 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], B, μa, Ta) 7. The natural extension of ([0, 1], B, μa, Ta) 8. The dynamical system ([0, 1], B, νa, Qa) 9. Generalized Hirzebruch-Jung continued fractions 10. The entropy of ([0, 1], B, θa, Ha) 11. The natural extension of ([0, 1], B, θa, Ha) 12. A new generalization of the Farey series Bibliography