The linear operators and ergodic theory of continued fractions with restricted partial quotients
by
Doug Hensley
Texas A & M University

Given a subset N of the positive integers, with two or more elements, there is a corresponding continued fraction Cantor set E_N. Given further a parameter \beta > 0 so that \sum_{n in N}n^-\beta is finite, there is an associated linear operator G=G_{\beta, N} acting on suitably defined spaces of functions, given by G[f](z):=\sum_{n in N}(n+z)^-\betaf(1/(n+z)). One of the results is that in general, at least if \beta > 1 (which includes the most inter esting cases) G is nuclear of order zero, with real eigenvalues, and with a positive dominant eigenvalue g=g_{\beta, N}.

The linear functional taking f to the coefficient of g in the natural decomposition of f into a sum of eigenfunctions is determined explicit ly by an integral of f over E_N with respect to a certain measure. This meas ure \mu on E_N is the natural measure in a sense; it reduces to Lebesgue mea sure when N={1, 2, 3, 4, ...}. A second measure \nu on E_N is the natural analogue of the Gauss measure of density 1/((1+t)log2. This gives entree to the ergodic theory of continued fractions with restricted partial quotients.

Date received: March 27, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-58.