Non-escaping endpoints do not explode

Abstract

The family of exponential maps $f_a(z)=e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\in (-\infty ,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$, it is known that $J(f_a)$ can be written as a union of "hairs" and "endpoints" of these hairs. In 1990, Mayer proved for $a\in (-\infty ,-1)$ that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where $a\in F(f_a)$, and showed that it holds even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a `spider's web'; in particular we give a new topological characterisation of spiders' webs that may be of independent interest. We also show how our results can be applied to Fatou's function, $z\mapsto z+1+e^{-z}$.

arXiv comment19 pages, 3 figures. v2: Final accepted manuscript, to appear in Bull. London Math. Soc

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Cited by in corpus (4)

• The topological dimension of radial Julia sets
• Dynamical sets whose union with infinity is connected
• The escaping set in transcendental dynamics
• Spiders' webs in the Eremenko-Lyubich class