Dynamic rays of bounded-type entire functions

By Günter Rottenfusser, Johannes Ruckert, Lasse Rempe, Dierk Schleicher. 2007

159 citations recorded by the source corpus.

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 nite order, every escaping point can be connected to 1 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ef41b5c3e046ca1da45b433dea37f1755b086559

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Walter Bergweiler
2. G. Stallard
3. P. Rippon
4. Lasse Rempe
5. Christopher J. Bishop
6. Dierk Schleicher

In Geometry and Topology

1. Walter Bergweiler
2. Lasse Rempe
3. G. Stallard
4. P. Rippon
5. Christopher J. Bishop
6. Dierk Schleicher

In Mathematical Physics

1. Walter Bergweiler
2. G. Stallard
3. P. Rippon
4. Lasse Rempe
5. Christopher J. Bishop
6. Dierk Schleicher

Citations in the corpus

• Fast escaping points of entire functions (2010, background): “For many functions in the class B, the escaping set consists of a family of curves to infinity, some with and some without endpoints (see [6, 17, 51]), and A(f) consists of these same curves excluding some of the endpoints (see [43]).”
• Constructing entire functions by quasiconformal folding (2015, methodology): “Note that, as above, no new dynamical argument is needed; we simply apply our S approximation results to the dynamical model constructed in [43].”
• Rigidity of escaping dynamics for transcendental entire functions (2006, methodology): “Remark. In [ RRRS ], the class Blog is defined without requiring condition (C).”
• Hyperbolic entire functions with bounded Fatou components (2014, unknown): “…it turns out that, even restricted to the hyperbolic case, entire functions can be incredibly diverse: for example, while for many such maps, the Julia sets are known to contain curves along which the iterates tend to infinity, there are also (hyperbolic) examples where this is not the case [51].”
• Multiply connected wandering domains of entire functions (2011, background): “Much work on the dynamics of entire functions has been concerned with the Speiser class and the Eremenko–Lyubich class, an important tool being a logarithmic change of variable which yields good distortion estimates; see [9, 23, 39, 45, 46] for some results concerning the Eremenko–Lyubich class.”
• Dynamics of meromorphic functions with direct or logarithmic singularities (2007, background): “D. Schleicher and J. Zimmer, ‘Escaping points of exponential maps’, J. London Math.”