Generalization of a theorem of Kusmin
A generalization of the Gauss-Kuzmin theorem in the metrical theory of continued fractions is given. The generalization is to consider instead of the map of the unit-interval to itself defined by the transformation \(Tx= \sum(k^{-1}- (k+x)^{-1})\), the map of the unit-interval to itself by the transformation \(T(x,y)= (Tx, x/(1- xTx+ xy))\). The main result of the paper is that the probability density function of the above transformation is \((\log 2)^{-1} (1+ xy)^{-2}\).
scientific article; zbMATH DE number 650849
scientific article