Convergence of continued fraction type algorithms and generators

Abstract

The concept of convergence of continued fraction type algorithms has been defined a number of times in the literature. We investigate the relation between these definitions, and show that they do not always coincide. We relate the definitions to the question whether or not the natural partition of the underlying dynamical system is a generator. It turns out that the ‘right’ definition of convergence is equivalent to this partition being a generator. The second definition of convergence is shown to be equivalent only under extra conditions on the transformation. These extra conditions are typically found to be satisfied when the second definition is used in the literature.