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The $\omega $-limit sets of quadratic Julia sets

Published online by Cambridge University Press:  27 September 2013

ANDREW D. BARWELL  and
BRIAN E. RAINES
ANDREW D. BARWELL
Affiliation:
Heilbronn Institute of Mathematical Research, University of Bristol, Howard House, Queens Avenue, Bristol BS8 1SN, UK email A.Barwell@bristol.ac.uk School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK email A.Barwell@bristol.ac.uk
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA email brian_raines@baylor.edu
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Abstract

In this paper we characterize $\omega $-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin’s symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an $\omega $-limit set of a point if, and only if, it is internally chain transitive.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 2 , April 2015 , pp. 337 - 358
Copyright
© Cambridge University Press, 2013 

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