Philip J. Rippon

Are Devaney hairs fast escaping

Beginning with Devaney, several authors have studied transcendental entire functions for which every point in the escaping set can be connected to infinity by a curve in the escaping set. Such curves are often called Devaney hairs. We show that, in many cases, every point in such a curve, apart from possibly a finite endpoint of the curve, belongs to the fast escaping set. We also give an example of a Devaney hair which lies in a logarithmic tract of a transcendental entire function and contains no fast escaping points.

Boundaries of univalent Baker domains

Let f be a transcendental entire function and let U be a univalent Baker domain of f. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of U form a set of harmonic measure zero with respect to U. This leads to a new sufficient condition for the escaping set of f to be connected, and also a new general result on Eremenko’s conjecture.

Exotic baker and wandering domains for Ahlfors islands maps

AbstractLet X be a compact Riemann surface of genus at most 1, i.e., the Riemann sphere or a torus, and let W ⊊ X be an arbitrary domain. We construct a variety of examples of holomorphic functions g: W → X that satisfy Epstein’s Ahlfors islands property and that have “pathological” dynamical behaviour. In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the ω-limit set of a Baker domain of such a function. We furthermore construct Ahlfors islands maps with wandering domains having prescribed ω-limit setswith logarithmic singularities having prescribed asymptotic curves and also produce examples where X is a compact hyperbolic surface. As a corollary of our method, we construct transcendental entire functions with Baker domains in which the iterates tend to infinity arbitrarily slowly.

Extensions of a Theorem of Valiron

A classical theorem of Valiron states that a function which is holomorphic in the unit disk, unbounded, and bounded on a spiral that accumulates at all points of the unit circle, has asymptotic value infinity. This property, and various other properties of such functions, are shown to hold for more general classes of functions which are bounded on a subset of the disk that has a suitably large set of nontangential limit points on the unit circle.

Transcendental Dynamics and Complex Analysis: Contents

Introduction 1. Iteration of inner functions and boundaries of components of the Fatou set D. Bargmann 2. Conformal automorphisms of finitely connected regions A. F. Beardon and D. Minda 3. Meromorphic functions with two completely invariant domains W. Bergweiler and A. Eremenko 4. A family of matings between transcendental entire functions and a Fuchsian group S. Bullett and M. Freiberger 5. Singular perturbations of z^n R. Devaney, M. Holzer, D. Look, M. Moreno Rocha and D. Uminsky 6. Residual Julia sets of rational and transcendental functions P. Dominguez and N. Fagella 7. Bank-Laine functions via quasiconformal surgery D. Drasin and J. K. Langley 8. Generalisations of uniformly normal families W. K. Hayman and A. Hinkkanen 9. Entire functions with bounded Fatou components A. Hinkkanen 10. On multiply connected wandering domains of entire functions M. Kisaka and M. Shishikura 11. Fractal measures and ergodic theory of transcendental meromorphic functions J. Kotus and M. Urbanski 12. Combinatorics of bifurcations in exponential parameter space L. Rempe and D. Schleicher 13. Baker domains P. J. Rippon 14. Escaping points of the cosine family G. Rottenfusser and D. Schleicher 15. Dimensions of Julia sets of transcendental meromorphic functions G. M. Stallard 16. Abels functional equation and its role in the problem of croissance reguliere G. Szekeres.

Transcendental dynamics and complex analysis : a tribute to Noel Baker

Introduction 1. Iteration of inner functions and boundaries of components of the Fatou set D. Bargmann 2. Conformal automorphisms of finitely connected regions A. F. Beardon and D. Minda 3. Meromorphic functions with two completely invariant domains W. Bergweiler and A. Eremenko 4. A family of matings between transcendental entire functions and a Fuchsian group S. Bullett and M. Freiberger 5. Singular perturbations of z^n R. Devaney, M. Holzer, D. Look, M. Moreno Rocha and D. Uminsky 6. Residual Julia sets of rational and transcendental functions P. Dominguez and N. Fagella 7. Bank-Laine functions via quasiconformal surgery D. Drasin and J. K. Langley 8. Generalisations of uniformly normal families W. K. Hayman and A. Hinkkanen 9. Entire functions with bounded Fatou components A. Hinkkanen 10. On multiply connected wandering domains of entire functions M. Kisaka and M. Shishikura 11. Fractal measures and ergodic theory of transcendental meromorphic functions J. Kotus and M. Urbanski 12. Combinatorics of bifurcations in exponential parameter space L. Rempe and D. Schleicher 13. Baker domains P. J. Rippon 14. Escaping points of the cosine family G. Rottenfusser and D. Schleicher 15. Dimensions of Julia sets of transcendental meromorphic functions G. M. Stallard 16. Abels functional equation and its role in the problem of croissance reguliere G. Szekeres.

Transcendental Dynamics and Complex Analysis: Frontmatter

Introduction 1. Iteration of inner functions and boundaries of components of the Fatou set D. Bargmann 2. Conformal automorphisms of finitely connected regions A. F. Beardon and D. Minda 3. Meromorphic functions with two completely invariant domains W. Bergweiler and A. Eremenko 4. A family of matings between transcendental entire functions and a Fuchsian group S. Bullett and M. Freiberger 5. Singular perturbations of z^n R. Devaney, M. Holzer, D. Look, M. Moreno Rocha and D. Uminsky 6. Residual Julia sets of rational and transcendental functions P. Dominguez and N. Fagella 7. Bank-Laine functions via quasiconformal surgery D. Drasin and J. K. Langley 8. Generalisations of uniformly normal families W. K. Hayman and A. Hinkkanen 9. Entire functions with bounded Fatou components A. Hinkkanen 10. On multiply connected wandering domains of entire functions M. Kisaka and M. Shishikura 11. Fractal measures and ergodic theory of transcendental meromorphic functions J. Kotus and M. Urbanski 12. Combinatorics of bifurcations in exponential parameter space L. Rempe and D. Schleicher 13. Baker domains P. J. Rippon 14. Escaping points of the cosine family G. Rottenfusser and D. Schleicher 15. Dimensions of Julia sets of transcendental meromorphic functions G. M. Stallard 16. Abels functional equation and its role in the problem of croissance reguliere G. Szekeres.