Using the methods of the noncommutative geometry and the K–theory, we prove that the well–known Dixmier–Douady invariant of continuous–trace C –algebras and the Godbillon– Vey invariant of the codimension–1 foliations on compact manifolds coincide in a class of the so–called ”foliation derived” C–algebra bundles. Moreover, with the help of such bundles both of the above invariants admit an eleg...

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...

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 di erent from the original one, given by Douady and Hubbard and re ned by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic depende...

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), Tn and Up(H), the Banach Lie group of unitaries differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and cal...

For an infinitely renormalizable quadratic map fc : z 7→ z 2+c with the sequence of renormalization periods {nm} and the rotation numbers {tm = pm/qm}, we prove that if lim supn−1 m log |pm| > 0, then the Mandelbrot set is locally connected at c. We prove also that if lim sup |tm+1| 1/qm < 1 and qm → ∞, then the Julia set of fc is not locally connected provided c is the limit of corresponding c...

1. The Parameter Space for Cubic Maps 2. Real Cubic Maps as Real Dynamical Systems 3. Complex Cubics: The Connectedness Locus 4. Hyperbolic Components Appendix A. Normal Forms and Curves in Parameter Space Appendix B. Centers of Some Hyperbolic Components Appendix C. Comments on the Figures References This article discusses the dynamics of iterated cubic maps from the real or complex line to it...

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this a...

An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable in H(X;Z)∪H(X;Z) ⊂ H(X;Z). One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic index of a projective family of elliptic operators...

A hypothesis is introduced under which a compact complex analytic space, X, viewed as a structure in the language of analytic sets, is essentially saturated. It is shown that this condition is met exactly when the irreducible components of the restricted Douady spaces of all the cartesian powers of X are compact. Some implications of saturation on Kähler-type spaces, which by a theorem of Fujik...