A two-dimensional Gauss-Kuzmin theorem for singular continued fractions

A very simple proof of a generalization of the Gauss-Kuzmin theorem for singular continued fractions is given by considering the transition operator defined in Se] as an operator on the Banach space BV(W) of complex-valued functions of bounded variation on W = 0, ( )]. The upper bound obtained here implies that the convergence rate, O(αⁿ), with 0.17 ≤ α ≤ 0.47 < g, is better than that obtained in DK].