# American Institute of Mathematical Sciences

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

 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.
Citation: William Ott, Qiudong Wang. Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1035-1054. doi: 10.3934/dcds.2010.26.1035
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