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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

Jawad Al-Khal 1, , Henk Bruin 2, and  Michael Jakobson 3,

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.
Keywords: infinite measure., sigma-finite invariant measure, interval maps.
Mathematics Subject Classification: Primary: 37E05; Secondary: 28D05, 37C4.
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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