Dimensions of Julia Sets of Hyperbolic Meromorphic Functions

It is known that, if f is a hyperbolic rational function, thenthe Hausdorff, packing and box dimensions of the Julia set,J(f), are equal. In this paper it is shown that, for a hyperbolictranscendental meromorphic function f, the packing and upperbox dimensions of J(f) are equal, but can be strictly greaterthan the Hausdorff dimension of J(f). 2000 Mathematics SubjectClassification 30D05, 37F10, 37F15, 37F35, 37F50.     

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.     

Bathophenanthroline-ruthenium(II) complexes as non-radioactive labels for oligonucleotides which can be measured by time-resolved fluorescence techniques

The specific attachment of bathophenanthroline-ruthenium(II) complexes as non-radioactive label molecules to synthetically 5′-NH₂-modified oligonucleotides is described. After excitation by light pulses, the fluorescence of these labels can be measured by a time-resolved mode woth high sensitivity. No quenching takes place due to coupling of the Ru complexes to the DNA. Ru-complex-labelled oligonucleotides still hybridize specifically to complementary DNA sequences, and no quenching is observed in the course of the hybridization process.     

Hysteretical and spectral behaviour of bistable cleaved coupled cavity semiconductor diode lasers

Cleaved coupled-cavity GaAs/AlGaAs diode lasers have been fabricated which display optical bistability with wide hysteresis loops in their light current (L/I) characteristics. The single longitudinal mode output of these devices has been investigated under several operating conditions. The observed spectral changes associated with mode hopping, shifting and jumping across the band have been explained qualitatively.     

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.