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Connectedness of Julia Sets of Rational Functions

Connectedness of Julia Sets of Rational Functions For a polynomial P it is well known that its Julia set % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! ${\cal J_P}$ is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point z 0 via a preimage of z 0 or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Connectedness of Julia Sets of Rational Functions

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Publisher
Springer Journals
Copyright
Copyright © 2001 by Heldermann Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03320977
Publisher site
See Article on Publisher Site

Abstract

For a polynomial P it is well known that its Julia set % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! ${\cal J_P}$ is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point z 0 via a preimage of z 0 or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 7, 2013

References