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

Rational parameter rays of the Mandelbrot set

We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries about hyperbolic components of the Mandelbrot set. Along the way, we introduce partitions of dynamical and parameter planes which are of independent interest, and we interpret the Mandelbrot set as a symbolic parameter space of kneading sequences and internal addresses.Comment: 33 pages, 9 figure

The Highly Unusual Outgassing of Comet 103P/Hartley 2 from Narrowband Photometry and Imaging of the Coma

We report on photometry and imaging of Comet 103P/Hartley 2 obtained at Lowell Observatory from 1991 through 2011. We acquired photoelectric photometry on two nights in 1991, four nights in 1997/98, and 13 nights in 2010/11. We observed a strong secular decrease in water and all other observed species production in 2010/11 from the 1991 and 1997/98 levels. We see evidence for a strong asymmetry with respect to perihelion in the production rates of our usual bandpasses, with peak production occurring ~10 days post-perihelion and production rates considerably higher post-perihelion. The composition was "typical", in agreement with the findings of other investigators. We obtained imaging on 39 nights from 2010 July until 2011 January. We find that, after accounting for their varying parentage and lifetimes, the C2 and C3 coma morphology resemble the CN morphology we reported previously. These species exhibited an hourglass shape in October and November, and the morphology changed with rotation and evolved over time. The OH and NH coma morphology showed hints of an hourglass shape near the nucleus, but was also enhanced in the anti-sunward hemisphere. This tailward brightness enhancement did not vary significantly with rotation and evolved with the viewing geometry. We conclude that all five gas species likely originate from the same source regions on the nucleus, but that OH and NH were derived from small grains of water and ammonia ice that survived long enough to be affected by radiation pressure and driven in the anti-sunward direction. We detected the faint, sunward facing dust jet reported by other authors, and did not detect a corresponding gas feature. This jet varied little during a night but exhibited some variations from night to night, suggesting it is located near the total angular momentum vector.Comment: Accepted by Icarus; 20 pages of text (preprint style), 5 tables, 7 figure

Geologic transcript from Apollo 11 mission

Quick time transcript of lunar geology from Apollo 1