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.

42 pages, 9 figures. V2. General revisions and corrections. Definition of the class \B_{\log} harmonized with other papers; some results added/restated for future reference; details added in proofs of Theorem 6.3 and of Lemma A.3