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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Topological conditions for
the existence of absorbing Cantor sets


Author: Henk Bruin
Journal: Trans. Amer. Math. Soc. 350 (1998), 2229-2263
MSC (1991): Primary 58F13, 58F11
MathSciNet review: 1458316
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Abstract: This paper deals with strange attractors of S-unimodal maps $f$. It generalizes earlier results in the sense that very general topological conditions are given that either

i)
guarantee the existence of an absorbing Cantor set provided the critical point of $f$ is sufficiently degenerate, or
ii)
prohibit the existence of an absorbing Cantor set altogether.
As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for $f$.


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Additional Information

Henk Bruin
Affiliation: Department of Mathematics, Royal Institute of Technology, 10044 Stockholm, Sweden
Email: bruin@math.kth.se

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02109-6
PII: S 0002-9947(98)02109-6
Keywords: Attractors, unimodal maps, invariant measures
Received by editor(s): June 1, 1995
Additional Notes: Supported by the Netherlands Organization for Scientific Research (NWO). The research for this paper was carried out during the author’s stay at the University of Erlangen-Nürnberg.
Article copyright: © Copyright 1998 American Mathematical Society