Abstract. Let f be a transcendental entire function and let A(f ) denote the set of points that escape to infinity 'as fast as possible' under iteration. By writing A(f ) as a countable union of closed sets, called 'levels' of A(f ), we obtain a new understanding of the structure of this set. For example, we show that if U is a Fatou component in A(f ), then ∂U ⊂ A(f ) and this leads to significant new results and considerable improvements to existing results about A(f ). In particular, we study functions for which A(f ), and each of its levels, has the structure of an 'infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

Abstract. 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 component and I(f ) ∩ J(f ) contains continua. Moreover, under this hypothesis I(f ) ∩ J(f ) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f ) ∩ J(f ) is 2 and the Hausdorff dimension of J(f ) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. This method is also applied to complex differential equations.

Abstract. 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 in any multiply connected wandering domain U of f . By introducing a certain positive harmonic function h in U , related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large n, the image domains U n = f n (U ) contain large annuli, C n , and that the union of these annuli acts as an absorbing set for the iterates of f in U . Moreover, f behaves like a monomial within each of these annuli and the orbits of points in U settle in the long term at particular 'levels' within the annuli, determined by the function h. We also discuss the proximity of ∂U n and ∂C n for large n, and the connectivity properties of the components of U n \ C n . These properties are deduced from new results about the behaviour of entire functions that omit certain values in an annulus.

Abstract. 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 Baker wandering domain, we show that I(f ) is a connected unbounded set.

Abstract. We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates f n (z) escape, that is, tend to ∞, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which f n (z) tends to ∞ at a bounded rate, and establish the connections between these sets and the Julia set of f . To do this, we show that the iterates of f satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a quasi-nested wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.