The vapor-phase Raman spectra,Raman band contour analyses,Coriolis constants,and force constants of spherical-top molecules MX4 (M = Group IV element, X = F,Cl, Br,or I)

The Raman spectra of 21 tetrahalides of Group IV have been recorded in the vapor phase at temperatures between 20 and 420° C. The Raman shifts of the Q branches of each fundamental are reported. In agreement with the selection rules, the ν₁(a₁) fundamental of each molecule has no rotational structure. With two exceptions, the separation between the OP and RS branches of the ν₂(e) fundamental of each molecule has been measured and found to be in close agreement with that calculated from an extended form of the Placzek-Teller theory. Due to first-order Coriolis coupling, the OP, RS branch separations of the ν₃(t₂) and ν₄(t₂) fundamentals of each molecule are different from each other and from those of the ν₂(e) fundamental. It has been possible to determine Coriolis constants for most of the tetrahalides from these branch separations, using both a previously published procedure and also a new procedure in which analytical expressions for each subbranch are derived. There is close agreement between the zeta values so determined and the corresponding values determined from infrared band contour analyses in those cases where the latter are available. Force constants based on the fundamental frequencies and the zeta values are reported.     

On questions of Fatou and Eremenko

Let be a transcendental entire function and let be the set of points whose iterates under tend to infinity. We show that has at least one unbounded component. In the case that has a Baker wandering domain, we show that is a connected unbounded set.

     

Dynamics of meromorphic functions with direct or logarithmic singularities

Let f be a transcendental meromorphic function and denote byJ(f) the Julia set and by I(f) the escaping set. We show thatif f has a direct singularity over infinity, then I(f) has anunbounded component and I(f)J(f) contains continua. Moreover,under this hypothesis I(f)J(f) has an unbounded component ifand only if f has no Baker wandering domain. If f has a logarithmicsingularity over infinity, then the upper box dimension of I(f)J(f)is 2 and the Hausdorff dimension of J(f) is strictly greaterthan 1. The above theorems are deduced from more general resultsconcerning functions which have ‘direct or logarithmictracts’, but which need not be meromorphic in the plane.These results are obtained by using a generalization of Wiman–Valirontheory. This method is also applied to complex differentialequations.     

Minimal Fine Limits of Subharmonic Functions of Slow Growth

Two results are proved which show that a subharmonic functionon the unit disc which does not grow too quickly and which doesnot have asymptotic value at too many points, must have finiteminimal fine limits at boundary points forming a set of positivelinear measure. Similar methods are used to obtain an asymptoticPhragmén-Lindelöf theorem for subharmonic functions.These results generalize and improve on earlier results forholomorphic functions.     

Functions of small growth with no unbounded Fatou components

We prove a form of the cos πρ 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.     

Exotic baker and wandering domains for Ahlfors islands maps

Let X be a compact Riemann surface of genus at most 1, i.e., the Riemann sphere or a torus, and let W ⊊ X be an arbitrary domain. We construct a variety of examples of holomorphic functions g: W → X that satisfy Epstein’s Ahlfors islands property and that have “pathological” dynamical behaviour. In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the ω-limit set of a Baker domain of such a function. We furthermore construct Ahlfors islands maps

On multiply connected wandering domains of meromorphic functions

We describe conditions under which a multiply connected wanderingdomain of a transcendental meromorphic function with a finitenumber of poles must be a Baker wandering domain, and we discussthe possible eventual connectivity of Fatou components of transcendentalmeromorphic 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)V, then f maps each component of U onto a componentof the boundary of V in . We give examples which show that our results are sharp; for example,we prove that a multiply connected wandering domain can mapto a simply connected wandering domain, and vice versa.