389 research outputs found
Boundaries of univalent Baker domains
Let be a transcendental entire function and let be a univalent Baker domain of . We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of form a set of harmonic measure zero with respect to . This leads to a new sufficient condition for the escaping set of to be connected, and also a new general result on Eremenko's conjecture
Connectedness properties of the set where the iterates of an entire function are unbounded
We investigate the connectedness properties of the set I+(f) of points where the iterates of an entire function f are unbounded. In particular, we show that I+(f) is connected whenever iterates of the minimum modulus of f tend to тИЮ. For a general transcendental entire function f, we show that I+(f)тИк \{\infty\} is always connected and that, if I+(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded
The iterated minimum modulus and conjectures of Baker and Eremenko
In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the minimum modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function
Functions of small growth with no unbounded Fatou components
We prove a form of the theorem which gives strong estimates
for the minimum modulus of a transcendental entire function of order zero. We
also prove a generalisation of a result of Hinkkanen that gives a sufficient
condition for a transcendental entire function to have no unbounded Fatou
components. These two results enable us to show that there is a large class of
entire functions of order zero which have no unbounded Fatou components. On the
other hand we give examples which show that there are in fact functions of
order zero which not only fail to satisfy Hinkkanen's condition but also fail
to satisfy our more general condition. We also give a new regularity condition
that is sufficient to ensure that a transcendental entire function of order
less than 1/2 has no unbounded Fatou components. Finally, we observe that all
the conditions given here which guarantee that a transcendental entire function
has no unbounded Fatou components, also guarantee that the escaping set is
connected, thus answering a question of Eremenko for such functions
On multiply connected wandering domains of meromorphic functions
We describe conditions under which a multiply connected wandering domain of a
transcendental meromorphic function with a finite number of poles must be a
Baker wandering domain, and we discuss the possible eventual connectivity of
Fatou components of transcendental meromorphic functions. We also show that if
is meromorphic, is a bounded component of and is the
component of such that , then maps each component of
onto a component of the boundary of in \hat{\C}. We give
examples which show that our results are sharp; for example, we prove that a
multiply connected wandering domain can map to a simply connected wandering
domain, and vice versa.Comment: 18 pages. To be published in the Journal of the London Mathematical
Societ
Classifying simply connected wandering domains
While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable
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