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

Jawad Al-Khal; Henk Bruin; Michael Jakobson

doi:10.3934/dcds.2008.22.35

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2008, Volume 22, Issue 1&2: 35-61. Doi: 10.3934/dcds.2008.22.35

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

1.
Department of Mathematics, Howard University, DC
2.
Department of Mathematics, University of Surrey
3.
Department of Mathematics, University of MD, College Park, MD

Received: May 2007

Revised: September 2007

Published: March 2008

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Abstract

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.

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Mathematics Subject Classification: Primary: 37E05; Secondary: 28D05, 37C40.

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