Hyperbolic Components in Exponential Parameter Space

We discuss the space of complex exponential maps \Ek\colon z\mapsto e^{z}+\kappa. We prove that every hyperbolic component $W$ 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

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

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

Value distribution and potential theory

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