Topological dynamics of exponential maps on their escaping sets

By Lasse Rempe. Ergodic Theory and Dynamical Systems, 2003

64 citations recorded by the source corpus.

For the family of exponential maps $E_{\kappa}(z)=\exp(z)+\kappa$, we prove an analog of Böttcher's theorem by showing that any two exponential maps $E_{\kappa_1}$ and $E_{\kappa_2}$ are conjugate on suitable subsets of their escaping sets, and this conjugacy is quasiconformal. Furthermore, we prove that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's ‘pinched disk model’ for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping points. We also answer several questions about escaping endpoints of dynamic rays. In particular, we give a necessary and sufficient condition for the ray to be continuously differentiable in such a point, and show that escaping points can escape arbitrarily slowly. Furthermore, we show that the principle of topological renormalization is false for attracting exponential maps.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7fbcb5b7c2390eeb57f390556fe94eae1cc7da51

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In Applied Mathematics

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

In Mathematical Physics

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

In Statistical and Nonlinear Physics

1. G. Stallard
2. Dierk Schleicher
3. P. Rippon

Citations in the corpus

• Slow escaping points of meromorphic functions (2008, background): “The set I(f) consists of the open curves (without endpoints) together with some of their finite endpoints; see [21] and [24].”
• The dynamical fine structure of iterated cosine maps and a dimension paradox (2004, background): “ched disk” models [D] or some kinds of combinatorial or even topological renormalization (although non-continuity of any possible renormalization for exponential maps has recently been shown by Rempe [R], and this argument is likely to apply in our cases too). While we have shown that all dynamic ra”