2,970 research outputs found

    Hyperbolic Components in Exponential Parameter Space

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    We discuss the space of complex exponential maps \Ek\colon z\mapsto e^{z}+\kappa. We prove that every hyperbolic component WW has connected boundary, and there is a conformal isomorphism \Phi_W\colon W\to\half^- which extends to a homeomorphism of pairs \Phi_W\colon(\ovl W,W)\to(\ovl\half^-,\half^-). This solves a conjecture of Baker and Rippon, and of Eremenko and Lyubich, in the affirmative. We also prove a second conjecture of Eremenko and Lyubich.Comment: To appear in: Comptes Rendues Acad Sci Paris.-- Detailed description of results can be found in ArXiv math.DS/0311480.-- 6 pages, 1 figur

    Quasi-exactly solvable quartic: real algebraic spectral locus

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    We describe the real quasi-exactly solvable spectral locus of the PT-symmetric quartic using the Nevanlinna parametrization.Comment: 17 pages, 11 figure

    Meromorphic solutions of higher order Briot-Bouquet differential equations

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    For differential equations P(y(k),y)=0,P(y^{(k)},y)=0, where PP is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate

    Value distribution and potential theory

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    We describe some results of value distribution theory of holomorphic curves and quasiregular maps, which are obtained using potential theory. Among the results discussed are: extensions of Picard's theorems to quasiregular maps between Riemannian manifolds, a version of the Second Main Theorem of Nevanlinna for curves in projective space and non-linear divisors, description of extremal functions in Nevanlinna theory and results related to Cartan's 1928 conjecture on holomorphic curves in the unit disc omitting hyperplanes
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