## Institute for Mathematical Sciences

## Preprint ims07-05

** G\"unter Rottenfu{\ss}er, Johannes R\"uckert, Lasse Rempe and Dierk Schleicher**
*Dynamic rays of bounded-type entire functions*

Abstract: We construct an entire function in the Eremenko-Lyubich class $\B$ 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 $\B$, 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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