Dynamic rays of bounded-type transcendental self-maps of the punctured plane

Abstract

We study the escaping set of functions in the class ${\mathcal{B}}^{\ast }$, that is, holomorphic functions $f:{\mathbb{C}}^{\ast }\to {\mathbb{C}}^{\ast }$ for which both zero and infinity are essential singularities, and the set of singular values of $f$ is contained in a compact annulus of ${\mathbb{C}}^{\ast }$. For functions in the class ${\mathcal{B}}^{\ast }$, escaping points lie in their Julia set. If $f$ is a composition of finite order transcendental self-maps of ${\mathbb{C}}^{\ast }$ (and hence, in the class ${\mathcal{B}}^{\ast }$), then we show that every escaping point of $f$ can be connected to one of the essential singularities by a curve of points that escape uniformly. Moreover, for every essential itinerary $e\in \{0,\infty {\}}^{\mathbb{N}}$, we show that the escaping set of $f$ contains a Cantor bouquet of curves that accumulate to $\{0,\infty \}$ according to $e$ under iteration by $f$.

References in corpus (5)

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• On a question of Eremenko concerning escaping components of entire functions
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Cited by in corpus (3)

• The escaping set of transcendental self-maps of the punctured plane
• Escaping Fatou components of transcendental self-maps of the punctured plane
• A Converse Landing Theorem in Parameter Spaces