We studied the rational realization problem for sub-hyperbolic semi-rational branched coverings. By using the shielding ring lemma, we are able to give a direct proof of CJS’s Theorem following the lines of the proof of Thurston’s Theorem given in the paper of Douady-Hubbard.

For the family of exponential maps z 7→ exp(z) + κ, we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that g is a periodic external ray of an exponential map with nonescaping singular value. Then g lands at a repelling or parabolic periodic point. We also show that there are periodic external rays landing at all periodic points of such an exponentia...

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1. Background and notation 2. The dynamical dichotomy of Fatou and Julia 3. Periodic points 4. The consequences of Montel's Theorem 5. The Julia set is the closure of the set of repelling periodic points 6. Classical results concerning the Fatou set 7. Sullivan's classification of the Fatou set 8. A condition for expansion on the Julia set 9. The dynamics of polynomials 10. The Mandelbrot set a...

Motivated by Wigner's theorem, a canonical construction is described that produces an Atiyah-Singer Dirac operator with both unitary and anti-unitary symmetries. This includes the for KR-theory as special case, filling long-standing gap in literature. In order to make construction, orientifold Spin-c-structures are defined classified using semi-equivariant Dixmier-Douady theory, analogues of se...

1. Introduction. To determine cohomology groups of holomorphic vector bundles or more general coherent analytic sheaves on complex manifolds is very important in several complex variables and complex geometry. For example, Cartan-Serre's theorem B and Kodaira's vanishing theorem are fundamental respectively in the studies of two classes of complex manifolds: Stein manifolds and projective algeb...

Let A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in A∗∗p (∼= A∗∗/L∗∗) if pAxp ⊂ pAp and px∗xp ∈ pAp then xp ∈ Ap (∼= A/L). The proof goes by interpreting elements of A∗∗p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space F (p) = {φ ∈ A∗ : φ ≥ 0, φ(p) = ‖φ‖ ≤ 1} in the notions developed by ...

The simplest version of the Maskit-Klein combination theorems concerns the action of a free product of two finite subgroups of PSL(2,C) on the Riemann sphere Ĉ, when these subgroups have fundamental domains whose interiors together cover Ĉ. We prove an analogous combination theorem for covering correspondences of rational maps, making use of Douady and Hubbard’s Straightening Theorem for polyno...