On a Class of Transformations which have Unique Absolutely Continuous Invariant Measures

Abstract

A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures.For this class of functions, a centra] limit theorem holds for all real measurable functions.For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.1. Introduction.In [1], Lasota and Yorke established the existence of absolutely continuous invariant measures for the class of piecewise C2 point transformations (see Definition 1 below) t from an interval into itself and satisfying inix\dT/dx\ > 1, wherever the slope exists.In [2], it was shown that if a function in this class has n discontinuities, then there are at most n independent, absolutely continuous invariant measures under t.The purpose of this paper is to show that a certain subclass C of the aforementioned class of functions possesses unique absolutely continuous invariant measures.Functions in (2 are characterized by a cornmunication property on the intervals that form the partition of the function.Using the method of symbolic dynamics [3], this property establishes the existence of a dense orbit, from which uniqueness follows.In [9], it is shown that a central theorem holds for the C2 transformations of [1] for a class of real Holder functions.In 5, we obtain a central limit theorem for transformations in 6 in a simpler way, and the theorem holds for all real measurable functions.In 6, it is shown that if t G 6 is piecewise linear and has a fixed point, then the unique absolutely continuous measure invariant under r is piecewise constant.2. Uniqueness theorem.Denote by (,, || ||) the space of all integrable functions on the interval J = [a, b].Let m denote the Lebesgue measure on /.Let t: J -* J be a measurable nonsingular transformation.By "nonsingular"