The estimate for the Lebesgue measure of preimages of iterates of an interval map

Abstract

For an interval map T with two indifferent fixed points, we provide both asymptotic estimates and error bounds for the Lebesgue measure of sets of the form $T^{-n}[a,b]$, where $[a,b]\subset (0,1)$. Two concise proofs are presented. The first follows from Heersink's result on the Farey map directly. The second use techniques from infinite ergodic theory along with an effective version of Karamata's Tauberian theorem.