Maximally and non-maximally fast escaping points of transcendental entire functions

Abstract

We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest. It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

References in corpus (2)

• Connectedness properties of the set where the iterates of an entire function are bounded
• On fundamental loops and the fast escaping set

Cited by in corpus (10)

• The bungee set in quasiregular dynamics
• Dynamical sets whose union with infinity is connected
• Connectedness properties of the set where the iterates of an entire function are unbounded
• Variations on a theme of Hardy concerning the maximum modulus
• The escaping set in transcendental dynamics
• The maximum modulus set of a polynomial
• Dynamics of meromorphic functions outside a countable set of essential singularities
• Iterates of Meromorphic Functions on Escaping Fatou Components
• The iterated minimum modulus and conjectures of Baker and Eremenko
• On slow escaping and non-escaping points of quasimeromorphic mappings