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