PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 12, December 2013, Pages 4249–4260 S 0002-9939(2013)11676-X Article electronically published on August 6, 2013 STRONGER LASOTA-YORKE INEQUALITY FOR ONE-DIMENSIONAL PIECEWISE EXPANDING TRANSFORMATIONS PEYMAN ESLAMI AND PAWEL G ´ ORA (Communicated by Bryna Kra) Abstract. For a large class of piecewise expanding C 1,1 maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previ- ously known 2/ inf |τ ′ |. Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim’s for a class of W-shaped maps. An- other application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps. 1. Introduction The problem of stability in general and the stability of invariant measures in particular are one of the most important (and difficult) questions in dynamical sys- tems. Here, we are concerned with the stability of absolutely continuous invariant measures (acim-stability) for piecewise expanding maps of an interval. The general setting is as follows. Definition 1.1 (acim-stability). Given a family of maps {τ ǫ : X → X} ǫ≥0 with cor- responding invariant densities {f ǫ } ǫ≥0 , we say that τ 0 is acim-stable if lim ǫ→0 τ ǫ = τ 0 implies lim ǫ→0 f ǫ = f 0 . The limits are taken with respect to properly chosen metrics on the space of maps and densities, respectively. A relevant notion of closeness for maps under consideration is convergence in the Skorokhod metric (see Definition 4.4), and for the corresponding invariant densities, in this paper it is convergence in L 1 . Stability problems were investigated in a multitude of works; most relevant to our study are [6] and [7]. The main motivation for this work was to prove acim-stability for some W-shaped maps with slopes > 1 (by a slope we shall always mean the absolute value of the slope). A troublesome property of such maps is that they contain periodic turning points. Let us consider such a map W with a fixed turning point p 0 . This would not be a problem if |W ′ | > 2 (whenever the derivative exists). In fact, then the acim-stability of W follows directly from the results of [6]. However, if 1 < |W ′ |≤ 2 near p 0 , the standard procedure, which is to work with an iterate of W that has Received by the editors October 6, 2011 and, in revised form, January 7, 2012 and February 1, 2012. 2010 Mathematics Subject Classification. Primary 37A10, 37A05, 37E05. The first author was supported by the INdAM-COFUND Marie Curie Fellowship during the final stages of the preparation of this article. c 2013 American Mathematical Society Reverts to public domain 28 years from publication 4249