A family of entire functions with Baker domains

In his paper [The iteration of polynomials and transcendental entire functions. J. Aust. Math. Soc. (Series A) 30 (1981), 483–495], Baker proved that the function f defined by f(z) = z+(sin?z/?z)+c has a Baker domain for c sufficiently large. In this paper we use a novel method to prove that f has a Baker domain for all c>0. We also prove that there exists an open unbounded set contained in the Baker domain on which the orbits of points under f are asymptotically horizontal

Spiders' webs and locally connected Julia sets of transcendental entire functions

We show that, if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider's web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider's web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider's web.Comment: 17 pages. v2: some corrections and improvements of a minor nature; to appear in Ergodic Theory Dynam. System

Uniformly bounded components of normality

Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and $$B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in U}\log(|w|+3)},$$ where the supremum $\sup_{U}$ is taken over all components of $F(f)$. If $B_1(f)<\infty$ or $B_2(f)<\infty$, then we say $F(f)$ is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, $F(f)$ is (strongly) uniformly bounded.Comment: 17 pages, a revised version, to appear in Mathematical Proceedings Cambridge Philosophical Societ

Danksagung an Prof. Maya Ninomiya

二宮まや教授古稀・退職記念

DYNAMICAL CONVERGENCE OF A CERTAIN POLYNOMIAL FAMILY TO f a (z) = z + e z + a

Abstract. A transcendental entire function f a (z) = z + e z + a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials + a, which converges uniformly on compact sets to f a as d → ∞. We show its dynamical convergence under a certain assumption, even though f a has a Baker domain or a wandering domain. We also investigate the parameter spaces of f a and P a,d

Universal Mandelbrot Set as a Model of Phase Transition Theory

The study of Mandelbrot Sets (MS) is a promising new approach to the phase transition theory. We suggest two improvements which drastically simplify the construction of MS. They could be used to modify the existing computer programs so that they start building MS properly not only for the simplest families. This allows us to add one more parameter to the base function of MS and demonstrate that this is not enough to make the phase diagram connectedComment: 5 pages, 3 figure

W. Raabes „Pfisters Mühle“ : eine Interpretation aus der Erzählweise

二宮まや教授古稀・退職記念