Generalized continued fractions

Abstract

The invariant measures for the transformations x bar right arrow {phi(x)} on the unit interval are studied where (i) phi is an element of C-2, (ii) phi' < 0, (iii) phi(0+) = +infinity, phi(1)= 1, (iv) the map x bar right arrow {phi(x)} is eventually expansive, and (v) phi' and phi " satisfy some growth conditions. It is shown that the invariant ergodic measure rho(x) dx satisfies rho(0) = (1 - 1/phi'(1)) . rho(1). For phi(x) = 1/x(p), the limiting behavior of rho is investigated as p --> +infinity or p --> p(0) = 0.241485... (C) 2000 Elsevier Science Inc. All rights reserved. AMS classification. Primary 28C10; 58F11.