Generalized Notions of Continued Fractions, Ergody and Number Theoretic Applications: A Review The theory of continued fractions has been around for centuries, with notable mathematicians like Euclid, Aryabhata, Fibonacci, Bombelli, Wallis, Huygens, and Euler making significant contributions to its development over time. This book focuses on the intriguing connections between ergodic theory and number theory, which have been established since the 1950s. It explores various generalizations and extensions of classical continued fractions, including generalized Lehner, simple, and Hirzebruch-Jung continued fractions. The book is intended for graduate students and senior researchers, and it begins by deriving invariant ergodic measures for each of the underlying transformations on [0,1]. It shows that famous formulas dating back to Khintchine and Levy carry over to more general settings. Additionally, the entropy of the transformations is calculated, and the natural extensions of the dynamical systems to [0,1]2 are analyzed. The book is divided into several chapters, each focusing on a specific aspect of continued fractions and their applications.

''