Typical points for one-parameter families of piecewise expanding maps of the interval

Daniel Schnellmann

doi:10.3934/dcds.2011.31.877

Article Contents

2011, Volume 31, Issue 3: 877-911. Doi: 10.3934/dcds.2011.31.877

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Typical points for one-parameter families of piecewise expanding maps of the interval

Daniel Schnellmann^1,

1.
Ecole Normale Supérieure, Départment de mathématiques et applications (DMA), 45 rue d’Ulm 75230 Paris cedex 05

Received: February 28, 2010

Revised: June 30, 2011

Published: August 2011

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Abstract

For one-parameter families of piecewise expanding maps of the interval we establish sufficient conditions such that a given point in the interval is typical for the absolutely continuous invariant measure for a full Lebesgue measure set of parameters. In particular, we consider $C^{1,1}(L)$-versions of $\beta$-transformations, piecewise expanding unimodal maps, and Markov structure preserving one-parameter families. For families of piecewise expanding unimodal maps we show that the turning point is almost surely typical whenever the family is transversal.

Keywords:

Typical points,
piecewise expanding maps of the interval,
absolutely continuous invariant measure.

Mathematics Subject Classification: Primary: 37E05, 37A05; Secondary: 37D20.

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References

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