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Item NEW EXAMPLES OF S-UNIMODAL MAPS WITH A SIGMA-FINITE ABSOLUTELY CONTINUOUS INVARIANT MEASURE(2004-12-09) Al-Khal, Jawad Yusuf; Jakobson, Michael V; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)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.