Dynamic rays of bounded-type entire functions

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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Authors

Günter Rottenfusser

Görresstraße 20
80798 München
Germany

Johannes Rückert

Jacobs University Bremen
28759 Bremen
Germany

Lasse Rempe

University of Liverpool
Liverpool L69 7ZL
United Kingdom

Dierk Schleicher

Jacobs University Bremen
28759 Bremen
Germany