466 research outputs found

    Connectedness properties of the set where the iterates of an entire function are unbounded

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    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

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    We prove a form of the cosπρ\cos \pi \rho 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

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    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 ff is meromorphic, UU is a bounded component of F(f)F(f) and VV is the component of F(f)F(f) such that f(U)Vf(U)\subset V, then ff maps each component of U\partial U onto a component of the boundary of VV 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

    Singularities of meromorphic functions with Baker domains

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    Boundaries of univalent Baker domains

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

    Slow escaping points of quasiregular mappings

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    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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