Dynamic rays of bounded-type entire functions

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.

💡 Research Summary

The paper addresses two long‑standing questions in complex dynamics: a problem posed by Eremenko in 1989 concerning whether every escaping point of an entire function in the Eremenko‑Lyubich class B can be joined to infinity by a curve of escaping points, and a question of Fatou from 1926 about the existence of such curves in general. The authors provide a striking dichotomy.

First, they construct an explicit entire function (f) belonging to class B whose Julia set consists solely of bounded path components. The construction uses a carefully designed infinite series of exponential terms with coefficients chosen so that each term dominates on a distinct annular “band” of the complex plane. This yields a function with bounded critical and asymptotic values (hence in class B) but whose escaping set (I(f)) is fragmented into a collection of compact “clusters”. Topological arguments show that none of these clusters contains an unbounded curve, thereby giving a negative answer to Eremenko’s question in full generality.

Second, the authors turn to subclasses of B where additional growth restrictions are imposed, notably functions of finite order (or more generally, functions whose maximum modulus grows at most like (\exp(r^{\rho})) for some finite (\rho)). By passing to logarithmic coordinates and exploiting the contraction properties of the inverse map on a sufficiently large exterior domain, they prove that every escaping point lies on a unique dynamic ray (also called a “hair”). These rays are pairwise disjoint, extend to infinity, and together form a “brush” that exhausts the entire escaping set. Consequently, for all functions in B of finite order, every escaping point can indeed be connected to infinity by a curve of escaping points, providing a positive partial answer to both Eremenko’s and Fatou’s questions.

The paper’s main theorems can be summarized as follows:

1. Negative example – There exists (f\in\mathcal{B}) such that every path component of its Julia set is bounded; consequently, (I(f)) contains no unbounded curves.

2. Positive result under growth control – If (f\in\mathcal{B}) has finite order (or satisfies a suitable finite‑type growth condition), then for each (z\in I(f)) there is a curve (\gamma_z\subset I(f)) joining (z) to (\infty). Moreover, the collection ({\gamma_z}) consists of pairwise disjoint dynamic rays that partition (I(f)).

These results illuminate the delicate interplay between the growth rate of an entire function and the topological structure of its escaping set. While the existence of dynamic rays is not guaranteed for all class B functions, it becomes inevitable once the function’s growth is sufficiently restrained. The authors conclude with several directions for future research, including the investigation of intermediate growth regimes, parameter space bifurcations of ray structures, and potential extensions to meromorphic functions with bounded singular sets.