On the approximation of invariant measures for continued fractions

Abstract

Kuzmin's theorem gives a sequence of functions which converge to the density of the invariant measure of n- dimensional continued fractions. The convergence is uniform and geometric. This paper gives bounds on the rate of con- vergence for two natural approximations to the sequence of functions given by Kuzmin's theorem. 1. Introduction. The metric theory of continued fractions, for n > 1, does not at this time include the form of the absolutely continuous in- variant measure for the associated shift transformation. Numerical as well as theoretical results have been obtained for this measure. In (l) some ergodic computations were performed in order to approximate the invariant measure for Jacobi's algorithm (the 2-dimen- sional continued fraction). Although an approximation was obtained, certain measure theoretic difficulties made an estimation of error in the approximation impossible. Thus it is of interest to find a technique where an estimate of the error is possible. After Schweiger proved the existence of the invariant measure, Kuzmin's theorem was proved for n > 1 in (6), Kuzmin's theorem, generalized to n-dimensions, gives a sequence of approximates to the density of the invariant measure which converges uniformly and geo- metrically. Many metric results, such as a geometric rate of mixing, follow from this theorem. The purpose of this paper is to give bounds on the rate of conver- gence of some natural approximations to the sequence of functions in Kuzmin's theorem, The evaluation of these functions involves summing over a countable set. To illustrate our techniques in a simpler setting and to provide some bounds for Gauss' measure, we deal with the one-dimensional case first. It should be pointed out that each of our approximation theorems depends on the measure of sets whose continued fraction expansions have bounded partial quotients.