Trees and the dynamics of polynomials

The basin of infinity of a polynomial map f : {\bf C} \arrow {\bf C} carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,\eta)$, with limiting dynamics F : T \arrow T. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary \PT_d compactifying the moduli space of polynomials of degree $d$. We show that $(T,\eta,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials \{f_t(z) : t \mem \Delta^* \} can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space \PT_3 is itself a tree. The metrized trees $(T,\eta,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\bf R}$-trees that arise as limits of hyperbolic manifolds.Comment: 60 page

Variation of solar-selective properties of black chrome with plating time

The spectral reflectance properties of a commercially prepared black chrome over dull nickel, both plated on steel, for various plating times of the black chrome were measured. The plating current was 180 amperes per square foot. Values of absorptance integrated over the solar spectrum, and of infrared emittance integrated over black-body radiation at 250 F were obtained. It is shown that plating between one and two minutes produces the optimum combination of highest heat absorbed and lowest heat lost by radiation

Holography Beyond the Penrose Limit

The flat pp-wave background geometry has been realized as a particular Penrose limit of AdS_5 x S^5. It describes a string that has been infinitely boosted along an equatorial null geodesic in the S^5 subspace. The string worldsheet Hamiltonian in this background is free. Finite boosts lead to curvature corrections that induce interacting perturbations of the string worldsheet Hamiltonian. We develop a systematic light-cone gauge quantization of the interacting worldsheet string theory and use it to obtain the interacting spectrum of the so-called `two-impurity' states of the string. The quantization is technically rather intricate and we provide a detailed account of the methods we use to extract explicit results. We give a systematic treatment of the fermionic states and are able to show that the spectrum possesses the proper extended supermultiplet structure (a non-trivial fact since half the supersymmetry is nonlinearly realized). We test holography by comparing the string energy spectrum with the scaling dimensions of corresponding gauge theory operators. We confirm earlier results that agreement obtains in low orders of perturbation theory, but breaks down at third order. The methods presented here can be used to explore these issues in a wider context than is specifically dealt with in this paper.Comment: v2: typo corrected in eqn. (6.2), version appearing in Nucl. Phys. B; LaTeX, 57 page

Exponential profiles from stellar scattering off interstellar clumps and holes in dwarf galaxy discs

Holes and clumps in the interstellar gas of dwarf irregular galaxies are gravitational scattering centers that heat field stars and change their radial and vertical distributions. Because the gas structures are extended and each stellar scattering is relatively weak, the stellar orbits remain nearly circular and the net effect accumulates slowly over time. We calculate the radial profile of scattered stars with an idealized model and find that it approaches an equilibrium shape that is exponential, similar to the observed shapes of galaxy discs. Our models treat only scattering and have no bars or spiral arms, so the results apply mostly to dwarf irregular galaxies where there are no other obvious scattering processes. Stellar scattering by gaseous perturbations slows down when the stellar population gets thicker than the gas layer. An accreting galaxy with a growing thin gas layer can form multiple stellar exponential profiles from the inside-out, preserving the remnants of each Gyr interval in a sequence of ever-lengthening and thinning stellar subdiscs.Comment: 10 pages, 5 figures, MNRAS accepte