Ergodic sums of non-integrable functions under one-dimensional dynamical systems with indifferent fixed points

We consider one-dimensional dynamical systems with indifferent fixed points (fixed points with derivative one). Many such maps have absolutely continuous ergodic infinite invariant measures. We study the limit of the ratio of the ergodic sum of f(A) to that of f(B), where the integrals of f(A) and f(B) are infinite with respect to the absolutely continuous ergodic infinite invariant measure. If f(A) and fB are analytic functions on [0,1], the result in this paper makes it clear whether the ratio of the ergodic sum of f(A) to that of f(B) converges in the Lebesgue measure or not.