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September  2010, 26(3): 1035-1054. doi: 10.3934/dcds.2010.26.1035

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, United States
2. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721-0089, United States

Received  January 2009 Revised  October 2009 Published  December 2009

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: nonuniformly expanding map, admissible family of $1D$ maps, parametrized family of maps, rank one map, absolutely continuous invariant measure., periodic attractor, singular limit.
Mathematics Subject Classification: Primary: 37D45, 37C4.
Citation: William Ott, Qiudong Wang. Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1035-1054. doi: 10.3934/dcds.2010.26.1035
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