NEW EXAMPLES OF S-UNIMODAL MAPS WITH A SIGMA-FINITE ABSOLUTELY CONTINUOUS INVARIANT MEASURE

Abstract

We construct new types of examples of S-unimodal maps &#38;#966; on an interval I that do not <br /> have a finite absolutely continuous invariant measure but that do have a &#38;#963; - finite one. <br /> These examples satisfy two important properties. The first property is topological, namely,<br /> the forward orbit of the critical point c is dense, i.e., &#38;#969;(c) = I. On the other hand, the <br /> second property is metric, we are able to conclude that this measure is infinite on every <br /> non-trivial interval. In the process, we show that we have the following dichotomy. <br /> Every absolutely continuous invariant measure, in our setting, is either &#38;#963; - finite, or else it <br /> is infinite on every set of positive Lebesgue measure. Our method of construction is based <br /> on the method of inducing a power map defined piecewise on a countable collection of <br /> non-overlapping intervals that partition I modulo a Cantor set of Lebesgue measure zero. <br /> The power map then satisfies what is known as the Folklore Theorem and therefore has <br /> a finite a.c.i.m. that is pulled back to define our &#38;#966; - invariant measure on I, with the above <br /> stated properties.