166 research outputs found

### 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

### 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

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