465 research outputs found
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 ∞. For a general transcendental entire function f, we show that I+(f)∪ \{\infty\} is always connected and that, if I+(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded
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
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
Slow escaping points of quasiregular mappings
This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f)
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