American Institute of Mathematical Sciences

January & February  2008, 22(1&2): 35-61. doi: 10.3934/dcds.2008.22.35

New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure

 1 Department of Mathematics, Howard University, DC, United States 2 Department of Mathematics, University of Surrey, United Kingdom 3 Department of Mathematics, University of MD, College Park, MD, United States

Received  May 2007 Revised  September 2007 Published  June 2008

We combine the technique of inducing with a method of Johnson boxes and construct new examples of S-unimodal maps $\varphi$ which do not have a finite absolutely continuous invariant measure, but do have a $\sigma$-finite one which is infinite on every non-trivial interval.
We prove the following dichotomy. Every absolutely continuous invariant measure is either $\sigma$-finite, or else it is infinite on every set of positive Lebesgue measure.
Citation: Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35
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