Dynamic rays of bounded-type entire functions

Günter Rottenfusser; Johannes Rückert; Lasse Rempe; Dierk Schleicher

doi:http://doi.org/10.4007/annals.2011.173.1.3

Abstract

We construct an entire function in the Eremenko-Lyubich class $\mathcal{B}$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\mathcal{B}$, in particular those of finite order, every escaping point can be connected to $\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.

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@article {dierkcosine, MRKEY = {2286634}, AUTHOR = {Schleicher, Dierk}, TITLE = {The dynamical fine structure of iterated cosine maps and a dimension paradox}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {136}, YEAR = {2007}, NUMBER = {2}, PAGES = {343--356}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {37F35 (37F10 37F20)}, MRNUMBER = {2286634}, MRREVIEWER = {Volker Mayer}, DOI = {10.1215/S0012-7094-07-13625-1}, ZBLNUMBER = {1114.37024}, }
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@article {dierkmonthly, MRKEY = {2321254}, AUTHOR = {Schleicher, Dierk}, TITLE = {Hausdorff dimension, its properties, and its surprises}, JOURNAL = {Amer. Math. Monthly}, FJOURNAL = {American Mathematical Monthly}, VOLUME = {114}, YEAR = {2007}, NUMBER = {6}, PAGES = {509--528}, ISSN = {0002-9890}, CODEN = {AMMYAE}, MRCLASS = {54F45 (28A78 28A80 37C45)}, MRNUMBER = {2321254}, MRREVIEWER = {Klaas Pieter Hart}, ZBLNUMBER = {1154.54023}, }
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@incollection{dierkentire, author = {Schleicher, Dierk}, TITLE = {Dynamics of entire functions}, BOOKTITLE={Proc. of the {\rm CIME} Summer School}, SERIES={Lecture Notes in Math.}, VOLUME={1998}, PAGES={295--339}, PUBLISHER={Springer-Verlag}, ADDRESS={New York}, YEAR={2010}, MRNUMBER={2648691}, DOI={10.1007/978-3-642-13171-4_5}, }
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@article {expescaping, MRKEY = {1956142}, AUTHOR = {Schleicher, Dierk and Zimmer, Johannes}, TITLE = {Escaping points of exponential maps}, JOURNAL = {J. London Math. Soc.}, FJOURNAL = {Journal of the London Mathematical Society. Second Series}, VOLUME = {67}, YEAR = {2003}, NUMBER = {2}, PAGES = {380--400}, ISSN = {0024-6107}, CODEN = {JLMSAK}, MRCLASS = {37F10 (30D05 39B12)}, MRNUMBER = {1956142}, MRREVIEWER = {Peter Ha{ï}ssinsky}, DOI = {10.1112/S0024610702003897}, ZBLNUMBER = {1042.30012}, }
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@misc{Sch07c, author={Schubert, H.}, TITLE={{Ü}ber die {H}ausdorff-{D}imension der {J}uliamenge von {F}unktionen endlicher {O}rdnung}, NOTE={doctoral dissertation, Christian-Albrechts-Universität Kiel}, YEAR={2007}, }
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@article {urbanskizdunik1, MRKEY = {1992945}, AUTHOR = {Urba{ń}ski, Mariusz and Zdunik, Anna}, TITLE = {The finer geometry and dynamics of the hyperbolic exponential family}, JOURNAL = {Michigan Math. J.}, FJOURNAL = {The Michigan Mathematical Journal}, VOLUME = {51}, YEAR = {2003}, NUMBER = {2}, PAGES = {227--250}, ISSN = {0026-2285}, MRCLASS = {37F10 (37A25 37F35 37F50)}, MRNUMBER = {1992945}, MRREVIEWER = {Rich L. Stankewitz}, DOI = {10.1307/mmj/1060013195}, ZBLNUMBER = {1038.37037}, }
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@article {zalcmannormal, MRKEY = {1624862}, AUTHOR = {Zalcman, Lawrence}, TITLE = {Normal families: new perspectives}, JOURNAL = {Bull. Amer. Math. Soc.}, FJOURNAL = {American Mathematical Society. Bulletin. New Series}, VOLUME = {35}, YEAR = {1998}, NUMBER = {3}, PAGES = {215--230}, ISSN = {0273-0979}, CODEN = {BAMOAD}, MRCLASS = {30D45 (34A20)}, MRNUMBER = {1624862}, MRREVIEWER = {Daniel Girela}, DOI = {10.1090/S0273-0979-98-00755-1}, ZBLNUMBER = {1037.30021}, }

Dynamic rays of bounded-type entire functions

Abstract

Authors