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Let f be a transcendental entire function and let I(f) be the set of points whose iterates under f tend to infinity. We show that I(f) has at least one unbounded component. In the case that f has a… (More)

Let f be a transcendental meromorphic function and denote by J(f) the Julia set and by I(f) the escaping set. We show that if f has a direct singularity over infinity, then I(f) has an unbounded… (More)

We look at the class Bn which contains those transcendental meromorphic functions f for which the finite singularities of f−n are in a bounded set and prove that, if f belongs to Bn, then there are… (More)

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than… (More)

Let $f$ be a transcendental entire function. The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping… (More)

Beginning with Devaney, several authors have studied transcen-dental entire functions for which every point in the escaping set can be connected to infinity by a curve in the escaping set. Such… (More)

We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates fn(z) escape, that is, tend to ∞, arbitrarily slowly. The proof uses new… (More)

The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function f… (More)