Dynamic rays of bounded-type entire functions

We construct an entire function in the Eremenko-Lyubich class $\text{B}$ 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 $\text{B}$, in particular those of finite order, every escaping point can be connected to $\mathrm{\infty }$ 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.