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Shadowing and $\omega $-limit sets of circular Julia sets

Published online by Cambridge University Press:  14 November 2013

ANDREW D. BARWELL ,
JONATHAN MEDDAUGH  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 [email protected] School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK email [email protected]
JONATHAN MEDDAUGH
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798–7328, USA email [email protected][email protected]
BRIAN E. RAINES
Affiliation:
Department of Mathematics, Baylor University, Waco, TX 76798–7328, USA email [email protected][email protected]
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Abstract

In this paper we consider quadratic polynomials on the complex plane ${f}_{c} (z)= {z}^{2} + c$ and their associated Julia sets, ${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an $n$-tupling. In this case ${J}_{c} $ contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that ${f}_{c} : {J}_{c} \rightarrow {J}_{c} $ has shadowing, and we classify all $\omega $-limit sets for these maps by showing that a closed set $R\subseteq {J}_{c} $ is internally chain transitive if, and only if, there is some $z\in {J}_{c} $ with $\omega (z)= R$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1045 - 1055
Copyright
© Cambridge University Press, 2013 

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