Continued fraction expansions with variable numerators

Abstract

A new continued fraction expansion algorithm, the so-called -expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada's -expansions, Schweiger's odd- and even-continued fraction expansions, and the Rosen fractions), these -expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the "shape" of the natural extension.