Periodic attractors versus nonuniform expansion  in singular limits of families of rank one maps

William Ott; Qiudong Wang

doi:10.3934/dcds.2010.26.1035

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2010, Volume 26, Issue 3: 1035-1054. Doi: 10.3934/dcds.2010.26.1035

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Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps

William Ott^1, and
Qiudong Wang^2,

1.
Department of Mathematics, University of Houston, Houston, TX 77204
2.
Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089

Received: December 31, 2008

Revised: September 30, 2009

Published: August 2010

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Abstract

We analyze parametrized families of multimodal $1D$ maps that arise as singular limits of parametrized families of rank one maps. For a generic $1$-parameter family of such maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure exhibits nonuniformly expanding dynamics characterized by the existence of a positive Lyapunov exponent and an absolutely continuous invariant measure. Under a mild combinatoric assumption, we prove that each such parameter is an accumulation point of the set of parameters admitting superstable periodic sinks.

Keywords:

parametrized family of maps,
rank one map,
singular limit,
admissible family of $1D$ maps,
periodic attractor,
nonuniformly expanding map,
absolutely continuous invariant measure.

Mathematics Subject Classification: Primary: 37D45, 37C40.

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