Recurrence of transformations with absolutely continuous invariant measures

Abstract

Let X = [0, 1]. If an ergodic transformation T : X-->X preserves an absolutely continuous probability measure rho(x) dx with rho(x) > 0, then it is shown that for almost every x is an element of X, [GRAPHICS] Define the kth first return time R-k(x) = min sless than or equal to1:\T(s)x-x\ less than or equal to 1/2(k)} and the kth recurrence error by epsilon(k) (x) = \T(R k (x))x-x\. Then it is shown that [GRAPHICS] (C) 2002 Elsevier Science Inc. All rights reserved.