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

William Ott, Qiudong Wang

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1035-1054
Number of pages20
JournalDiscrete and Continuous Dynamical Systems
Volume26
Issue number3
DOIs
StatePublished - Mar 1 2010

Keywords

  • Absolutely continuous invariant measure
  • Admissible family of 1d maps
  • Nonuniformly expanding map
  • Parametrized family of maps
  • Periodic attractor
  • Rank one map
  • Singular limit

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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