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

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

Only connect: addressing the emotional needs of Scotland's children and young people

A report on the SNAP (Scottish Needs Assessment Programme) Child and Adolescent Mental Health Phase Two survey. It describes a survey of a wide range of professionals working with children and young people in Scotland, and deals with professional perspectives on emotional, behavioural and psychological problems. Conclusions and recommendations are presented

Classifying simply connected wandering domains

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable

Variability of Jovian ion winds: an upper limit for enhanced Joule heating

It has been proposed that short-timescale fluctuations about the mean electric field can significantly increase the upper atmospheric energy inputs at Jupiter, which may help to explain the high observed thermospheric temperatures. We present data from the first attempt to detect such variations in the Jovian ionosphere. Line-of-sight ionospheric velocity profiles in the Southern Jovian auroral/polar region are shown, derived from the Doppler shifting of H<sub>3</sub><sup>+</sup> infrared emission spectra. These data were recently obtained from the high-resolution CSHELL spectrometer at the NASA Infrared Telescope Facility. We find that there is no variability within this data set on timescales of the order of one minute and spatial scales of 640 km, putting upper limits on the timescales of fluctuations that would be needed to enhance Joule heating