A+ CATEGORY SCIENTIFIC UNIT

Uncountable $\omega $-limit sets with isolated points

Volume 205 / 2009

Chris Good, Brian E. Raines, Rolf Suabedissen Fundamenta Mathematicae 205 (2009), 179-189 MSC: 37B45, 37E05, 54F15, 54H20. DOI: 10.4064/fm205-2-6

Abstract

We give two examples of tent maps with uncountable (as it happens, post-critical) $\omega $-limit sets, which have isolated points, with interesting structures. Such $\omega $-limit sets must be of the form $C\cup R$, where $C$ is a Cantor set and $R$ is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable $\omega $-limit sets for finite-to-one maps satisfying at least some weak form of expansivity. We show that this restriction does not hold if the $\omega $-limit set is uncountable. Secondly, we give an example of an $\omega $-limit set of the form $C\cup R$ for which the Cantor set $C$ is minimal.

Authors

  • Chris GoodSchool of Mathematics and Statistics
    University of Birmingham
    Birmingham, B15 2TT, UK
    e-mail
  • Brian E. RainesDepartment of Mathematics
    Baylor University
    Waco, TX 76798-7328, U.S.A.
    e-mail
  • Rolf SuabedissenMathematical Institute
    University of Oxford
    Oxford, OX1 3LB, UK
    e-mail

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