Topological Dynamics of Exponential Maps on their Escaping Sets

Abstract

We develop an abstract model for the dynamics of an exponential map $z\mapsto \exp (z)+\kappa$ on its set of escaping points and, as an analog of Boettcher's theorem for polynomials, show that every exponential map is conjugate, on a suitable subset of its set of escaping points, to a restriction of this model dynamics. Furthermore, we show that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's "pinched disk model" for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping sets. Using the correspondence with our model, we also answer several questions about escaping endpoints of external rays, such as when a ray is differentiable in such an endpoints or how slowly these endpoints can escape to infinity.

38 pages, 3 figures. // V3: Several typos fixed; some overall revision; parts of the material in Sections 5, 7 and 11 have been rewritten

References in corpus (2)

• Rational rays and critical portraits of complex polynomials
• A Landing Theorem for Periodic Rays of Exponential Maps

Cited by in corpus (28)

• Rigidity of escaping dynamics for transcendental entire functions
• Brushing the hairs of transcendental entire functions
• Classification of Escaping Exponential Maps
• Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds
• A Landing Theorem for Periodic Rays of Exponential Maps
• Escaping endpoints explode
• Connected escaping sets of exponential maps
• On Nonlanding Dynamic Rays of Exponential Maps
• Are Devaney hairs fast escaping?
• Foundations for an iteration theory of entire quasiregular maps
• A landing theorem for entire functions with bounded post-singular sets
• Geometrically finite transcendental entire functions
• Parameter rays for the exponential family
• The topological dimension of radial Julia sets
• Non-escaping endpoints do not explode
• Triviality of fibers for Misiurewicz parameters in the exponential family
• On the differentiability of hairs for Zorich maps
• On the connectivity of the escaping set for complex exponential Misiurewicz parameters
• The escaping set in transcendental dynamics
• Distinguishing endpoint sets from Erdős space
• On the boundary of an immediate attracting basin of a hyperbolic entire function
• Erdős space in Julia sets
• The Dynamical Fine Structure of Iterated Cosine Maps and a Dimension Paradox
• Indecomposable continua in exponential dynamics-Hausdorff dimension
• Infinite-dimensional Thurston theory and transcendental dynamics II: classification of entire functions with escaping singular orbits
• On the dimension of points which escape to infinity at given rate under exponential iteration
• Exponential iteration and Borel sets
• Hausdorff measure of hairs without endpoints in the exponential family