Fast escaping points of entire functions

Abstract

Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

References in corpus (2)

• Connected escaping sets of exponential maps
• On the connectivity of the escaping set for complex exponential Misiurewicz parameters

Cited by in corpus (48)

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• Maximally and non-maximally fast escaping points of transcendental entire functions
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• Superattracting fixed points of quasiregular mappings
• Hollow quasi-Fatou components of quasiregular maps
• The size and topology of quasi-Fatou components of quasiregular maps
• Geometrically finite transcendental entire functions
• Lebesgue measure of escaping sets of entire functions of completely regular growth
• On fundamental loops and the fast escaping set
• On the set where the iterates of an entire function are bounded
• The Julia set and the fast escaping set of a quasiregular mapping
• Poincaré functions with spiders' webs
• The exponential map is chaotic: An invitation to transcendental dynamics
• Non-escaping endpoints do not explode
• The topological dimension of radial Julia sets
• Dynamical sets whose union with infinity is connected
• The escaping set of transcendental self-maps of the punctured plane
• Eremenko's conjecture, wandering Lakes of Wada, and maverick points
• Connectedness properties of the set where the iterates of an entire function are unbounded
• Fast escaping points of entire functions: a new regularity condition
• Dimensions of slowly escaping sets and annular itineraries for exponential functions
• The dynamics of quasiregular maps of punctured space
• A note on the topology of escaping endpoints
• Lyapunov exponents and related concepts for entire functions
• A sharp growth condition for a fast escaping spider's web
• The escaping set in transcendental dynamics
• Baker's conjecture and Eremenko's conjecture for functions with negative real zeros
• Dynamics of meromorphic functions outside a countable set of essential singularities
• Spiders' webs in the Eremenko-Lyubich class
• Permutable Quasiregular Maps
• Hausdorff dimension in quasiregular dynamics
• A new geometric characterization of the Julia set
• Spiders' webs and locally connected Julia sets of transcendental entire functions
• Julia and escaping set spiders' webs of positive area
• On the dimension of points which escape to infinity at given rate under exponential iteration
• Fast Escaping Sets of meromorphic functions
• The iterated minimum modulus and conjectures of Baker and Eremenko
• A new family of entire functions with no wandering domains
• Iterates of Meromorphic Functions on Escaping Fatou Components
• Escaping points of commuting meromorphic functions with finitely many poles
• Fast Escaping Set of Transcendental Semigroup
• Spider's webs of doughnuts
• Area of the complement of the fast escaping sets of a family of entire functions
• Annular itineraries for entire functions
• Functions of genus zero for which the fast escaping set has Hausdorff dimension two
• Simply connected fast escaping Fatou components
• Iteration of quasiregular tangent functions in three dimensions