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 ∞ 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.
The points which converge to ∞ under iteration of the maps z −→ λ exp(z) for λ ∈ C\{0} are investigated. A complete classification of such 'escaping points' is given: they are organized in the form of differentiable curves called rays which are diffeomorphic to open intervals, together with the endpoints of certain (but not all) of these rays. Every escaping point is either on a ray or the endpoint (landing point) of a ray. This answers a special case of a question of Eremenko. The combinatorics of occurring rays, and which of them land at escaping points, are described exactly. It turns out that this answer does not depend on the parameter λ.It is also shown that the union of all the rays has Hausdorff dimension 1, while the endpoints alone have Hausdorff dimension 2. This generalizes results of Karpińska for specific choices of λ.
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