Conformal,geometric and invariant measures for transcendental expanding functions

We describe the fractal structure of expanding meromorphic maps of the form , where H and Q are rational functions whose most transparent examples are among the functions of the form with . In particular we show that depending upon whether the Hausdorff dimension of the Julia set is greater or less than 1, the finite non-zero geometric measure is provided by the Hausdorff or packing measure. In order to describe this fractal structure we introduce and explore in detail Walters expanding conformal maps and jump-like conformal maps. Received: 3 May 2001 / Published online: 5 September 2002     

On Transfer Operators for Continued Fractions with Restricted Digits

For any non-empty subset I of the natural numbers, let _I denotethose numbers in the unit interval whose continued fractiondigits all lie in I. Define the corresponding transfer operator for , where Re (rß) = _I is the abscissa of convergence of the series . When acting on a certain Hilbert space H_I, rß, weshow that the operator L_I, rß is conjugate to an integraloperator K_I, rß. If furthermore rß is real,then K_I, rß is selfadjoint, so that L_I, rß: H_I, rß H_I, rß has purely real spectrum.It is proved that L_I, rß also has purely real spectrumwhen acting on various Hilbert or Banach spaces of holomorphicfunctions, on the nuclear space C [0, 1], and on the Fréchetspace C [0, 1]. The analytic properties of the map rß L_I, rßare investigated. For certain alphabets I of an arithmetic nature(for example, I = primes, I = squares, I an arithmetic progression,I the set of sums of two squares it is shown that rß L_I, rß admits an analytic continuation beyond thehalf-plane Re rß > _I. 2000 Mathematics SubjectClassification 37D35, 37D20, 30B70.     

Conformal Families of Measures for Fibred Systems

In this paper, we give sufficient conditions for the existence of (pseudo) weakly conformal and conformal families of measures for fibred systems. We describe a general construction principle for these families, modelled on the one developed by Denker and Urbanski for conformal measures. For those systems that are fibrewise local homeomorphisms, the constructed families are (pseudo) conformal. If a system is, moreover, weakly topologically exact along fibres, then each measure in the associated family is supported on the whole fibre where it is naturally defined.     

Structural studies on some 1,3,5,7-tetraazabicyclo-[3.3.1]-nonane derivatives

From a study of the ¹H NMR spectra of a number of 3,7-disubstituted derivatives of 1,3,5,7-tetraazabicycio - [3.3.1] - nonane it is concluded that these molecules exist in either chair-chair or flattened chair-chair conformations. The derivatives containing COCH₃ and NO substitutents have room temperature spectra consistent with restricted rotation about their NC and NN bonds respectively. High temperature spectra reveal an activation energy, for the rotational barrier of the NC bond in the diacetyl derivative, of 23±2K.Cal/mol.     

How Points Escape to Infinity Under Exponential Maps

We investigate the finer fractal structure of the set of pointsescaping to infinity under iteration of an arbitrary exponentialmap. Providing exact formulas, we show how sensitively the Hausdorffdimension depends on the rate of growth of canonical Devaney–Krychcodes.     

Finer geometric rigidity of limit sets of conformal IFS

We consider infinite conformal iterated function systems in the phase space with . Let be the limit set of such a system. Under a mild technical assumption, which is always satisfied if the system is finite, we prove that either the Hausdorff dimension of exceeds the topological dimension of the closure of or else the closure of is a proper compact subset of either a geometric sphere or an affine subspace of dimension . A similar dichotomy holds for conformal expanding repellers.

     

On Invariant Line Fields

It is shown that a rational function of degree 2 admits aninvariant line field with respect to some measure µ, whichis an equilibrium state of a Hölder continuous potentialwhose topological pressure is greater than its supremum, onlyin very special cases when the Julia set is either a geometriccircle or an interval, or totally disconnected and containedin a real-analytic curve. 1991 Mathematics Subject Classification37F10, 30E99, 28D05, 28D99.     

Chemically and thermally stable,emissive carbon dots as viable alternatives to semiconductor quantum dots for emissive nematic liquid crystal–nanoparticle mixtures with lower threshold voltage

Dispersions of chemically and thermally robust carbon dots (2.5 ± 0.5 nm in core diameter) were prepared and investigated by polarised optical microscopy, electro-optic measurements including dynamic tests and numerical simulations as well as fluorescence confocal microscopy. The carbon dots were prepared by a straightforward thermal decomposition method from citric acid and hexadecylamine, and they show typical excitation wavelength-dependent photoluminescence behaviour. All dispersions, ranging from 0.5 to 5.0 wt.%, showed lower values for isotropic–nematic phase transition temperature and broader isotropic–nematic biphasic temperature intervals with increasing carbon dot content in comparison to the neat material. Doping of the nematic host with the carbon dots resulted in lower values for the apparent threshold voltage and the elastic constants, but higher values for the rotational viscosity. At 2.5 wt.% and higher, carbon dots residing at the confining interfaces in planar cells induce an increasing initial pre-tilt of up to 8° at lower temperatures. Fluorescence confocal microscopy confirmed this, where the luminescence of the carbon dots permitted visualisation of the distribution of the carbon dots in the bulk with a noticeable, in some cases even pattern-like, segregation to the confining interfaces.     

Aspects of nanometer scale imaging with extreme ultraviolet (EUV) laboratory sources

Imaging systems with nanometer resolution are instrumental to the development of the fast evolving field of nanoscience and nanotechnology. Decreasing the wavelength of illumination is a direct way to improve the spatial resolution in photon-based imaging systems and motivated a strong interest in short wavelength imaging techniques in the extreme ultraviolet (EUV) region. In this review paper, various EUV imaging techniques, such as 2D and 3D holography, EUV microscopy using Fresnel zone plates, EUV reconstruction of computer generated hologram (CGH) and generalized Talbot self-imaging will be presented utilizing both coherent and incoherent compact laboratory EUV sources. Some of the results lead to the imaging with spatial resolution reaching 50 nm in a very short exposure time. These techniques can be used in a variety of applications from actinic mask inspection in the EUV lithography, biological imaging to mask-less lithographic processes in nanofabrication.