Continued fractions and their generalizations. A short history of \(f\)-expansions (Q2813265)

From the publisher's description: ``This book is about the history of \(f\)-expansions, their theory, applications, and their connection to other parts of mathematics. Sketches of proofs of some of the theorems about \(f\)-expansions -- particularly theorems from historical sources -- are included not to convince the reader of the truth of the theorem but rather as a way to demonstrate why the theorem is true. These sketches should give a clearer and more easily understood description of the working of the theorem than a hand-waving literary flourish.''NEWLINENEWLINE NEWLINEThe book contains 22 short chapters and a list of sources.NEWLINENEWLINENEWLINEAfter a short introduction, elements of the theory of continued fractions are presented in the second chapter. Expansions into a series and infinite products are contained in the third and fourth chapters. Kakeya's results form the fifth chapter. The cotangent algorithm is discussed in the sixth chapter. The main ideas of papers of Bissinger and Everett are given in the seventh chapter. The Borel-Bernstein theorem is contained in the eighth chapter. The ninth chapter is called ``Measure zero and one''. The results that have their origins in a paper of Rényi are presented in the tenth chapter. The eleventh chapter is called ``Gauss, Kuzmin, and their followers'' and the twelfth chapter is called ``Lévy and the dual algorithm''. The \(\beta\)-expansions are presented in thirteenth chapter. The Japanese continued fractions are discussed in the fourteenth chapter. The discontinuous groups are described in the fifteenth chapter. The next, the sixteenth chapter is called ``Ergodic map''. Invariant measures are described in the seventeenth chapter and further ideas on invariant measures are discussed in the next chapter. The nineteenth chapter is called ``Entropy''. The Hausdorff dimension is described in twentieth chapter. The twenty-first chapter is called ``Multidimensional generalizations''. The theory of chaos is discussed in the twenty-second chapter.