Multiply connected wandering domains of entire functions

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 $\partial U_n$ and $\partial C_n$ for large $n$, and the connectivity properties of the components of $U_n \setminus \bar{C_n}$. These properties are deduced from new results about the behaviour of an entire function which omits certain values in an annulus

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

Functions of small growth with no unbounded Fatou components

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

Traces, CSLBS Newsletter Spring 2023

With contributions from Matthew J. Smith, Isaac Crichlow, and Matthew Stallard

Traces, CSLBS Newsletter Autumn 2022

With contributions from Matthew J. Smith, Matthew Stallard, and Jess Hannah

Operational loads on a tidal turbine due to environmental conditions

Accurate assessment of the fatigue life of tidal stream turbines and components requires prediction of the unsteady loading of turbine components over a wide range of frequencies. This study focuses on the influence of ambient turbulence, velocity shear and the approach taken to model wave kinematics, on the variation of thrust load imposed on the rotor shaft and supporting tower. Load cycles are assessed based on sea-state occurrence data taken over a five month period for a case study site. The influence of each environmental parameter on component loading is evaluated and the impact on material design parameters assessed. Alternative approaches are considered for modelling turbulent loading and wave loading. The frequency variation of loads due to turbulence are scaled from experimental data from trials of a three-bladed horizontal axis turbine of 1.2 m diameter on a bed-mounted supporting structure. Frequency dependent wave loading is estimated by a relative form of the drag term of the widely used equation of Morison et al. (1950), with the depth decay of kinematics modelled by linear wave theory. Over the five month interval considered a ten year design life can be obtained with a lower design load by accounting for variation of turbulence intensity that occurs during each tidal cycle. This is expected to vary further with the approach taken to model the onset turbulence. A component can also be designed for lower loads over the same time period if irregular waves are modelled instead of regular