Spiders' webs in the Eremenko-Lyubich class

Abstract

Consider the entire function $f(z)=\cosh (z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture of Sixsmith from 2020. In fact, we show that the "fast escaping set", i.e. the set of points whose orbits tend to infinity at an iterated exponential rate, is a spider's web. This answers a question of Rippon and Stallard from 2012. We also discuss a wider class of functions to which our results apply, and state some open questions.

10 pages, to appear in Int. Math. Res. Not. V3: Author accepted manuscript; some clarifications and improvements to the exposition throughout

References in corpus (11)

• Iteration of meromorphic functions
• Fast escaping points of entire functions
• Rigidity of escaping dynamics for transcendental entire functions
• Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
• Connected escaping sets of exponential maps
• Are Devaney hairs fast escaping?
• Geometrically finite transcendental entire functions
• Orbifold expansion and entire functions with bounded Fatou components
• Non-escaping endpoints do not explode
• The Eremenko-Lyubich constant
• Dynamical sets whose union with infinity is connected

Cited by in corpus (1)

• The escaping set in transcendental dynamics