Generalized Notions of Continued Fractions: Ergodicity and Number Theoretic Applications

Ancient times saw the birth of the theory of continued fractions, which has been developed by mathematical geniuses such as Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, and Euler. This book examines several generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions, and calculates the entropy of the transformations. It is suitable for graduate students and senior researchers and contains the basic background needed to understand the topic.

Format: Hardback
Length: 142 pages
Publication date: 20 July 2023
Publisher: Taylor & Francis Ltd

Ancient times saw the birth of the theory of continued fractions, a mathematical concept that has been nurtured by the brilliance of mathematical geniuses like Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, and Euler over centuries. This renowned theory continues to evolve, serving as a powerful bridge between diverse branches of mathematics.

This comprehensive book, aimed primarily at graduate students and seasoned researchers, delves into the captivating interplay between ergodic theory and number theory, a relationship that has flourished since the 1950s. It explores various generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. By deriving invariant ergodic measures for these underlying transformations on the interval [0, 1], the book demonstrates that renowned formulas dating back to Khintchine and Levy hold true in more general settings. Additionally, it calculates the entropy of these transformations and examines the natural extensions of dynamical systems to [0, 1]2.

With its clear and concise writing style, this book is an invaluable resource for advanced students and researchers in number theory. It provides a solid foundation, including essential background material, making it a self-contained volume that can be approached independently. By unraveling the intricate connections between ergodic theory and continued fractions, this book offers a deeper understanding of these fundamental mathematical concepts and their applications.

Dimension: 234 x 156 (mm)
ISBN-13: 9781032516783