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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TITLE = {On the dynamical fine structure of entire transcendental functions},
SCHOOL={International University Bremen},
YEAR={2005},
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AUTHOR = {Rottenfusser, G{ü}nter and Schleicher, Dierk},
TITLE = {Escaping points of the cosine family},
BOOKTITLE = {Transcendental {{D}}ynamics and {{C}}omplex {{A}}nalysis},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {348},
PAGES = {396--424},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2008},
MRCLASS = {37F10 (30D05 37F20 37F35)},
MRNUMBER = {2458810},
MRREVIEWER = {Walter Bergweiler},
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ISSN = {0012-7094},
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}