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Conformal Geometry and Dynamics

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Limit functions of discrete dynamical systems


Authors: H.-P. Beise, T. Meyrath and J. Müller
Journal: Conform. Geom. Dyn. 18 (2014), 56-64
MSC (2010): Primary 37A25, 37F10, 30K99
DOI: https://doi.org/10.1090/S1088-4173-2014-00264-1
Published electronically: April 1, 2014
MathSciNet review: 3187620
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Abstract: In the theory of dynamical systems, the notion of $ \omega $-limit sets of points is classical. In this paper, the existence of limit functions on subsets of the underlying space is treated. It is shown that in the case of topologically mixing systems on appropriate metric spaces $ (X,d)$, the existence of at least one limit function on a compact subset $ A$ of $ X$ implies the existence of plenty of them on many supersets of $ A$. On the other hand, such sets necessarily have to be small in various respects. The results for general discrete systems are applied in the case of Julia sets of rational functions and in particular in the case of the existence of Siegel disks.


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Additional Information

H.-P. Beise
Affiliation: Department of Mathematics, University of Trier, 54286 Trier, Germany
Email: pbeise@gmx.de

T. Meyrath
Affiliation: University of Luxembourg, Faculte des Sciences 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg
Email: thierry.meyrath@uni.lu

J. Müller
Affiliation: University of Trier, Mathematik, Fachbereich IV, 54286 Trier, Germany
Email: jmueller@uni-trier.de

DOI: https://doi.org/10.1090/S1088-4173-2014-00264-1
Keywords: Julia set, limit set, Siegel disk, universality
Received by editor(s): May 22, 2013
Received by editor(s) in revised form: November 20, 2013, December 30, 2013, and December 31, 2013
Published electronically: April 1, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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