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

摘要

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 f_B 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.