Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.

The IMS preprints are also available from the mathematics section of the arXiv e-print server, which offers them in several additional formats. To find the IMS preprints at the arXiv, search for Stony Brook IMS in the report name.

**PREPRINTS IN THIS SERIES, IN PDF FORMAT.**

* Starred papers have appeared in the journal cited.

*A Remark on Herman's Theorem for Circle Diffeomorphisms*

We define a class of real numbers that has full measure and is contained in the set of Roth numbers. We prove the $C^1$ - part of Herman's theorem: if f is a $C^3$ diffeomorphism of the circle to itself with a rotation number ω in this class, then f is $C^1$ --conjugate to a rotation by ω. As a result of restricting the class of admissible rotation numbers, our proof is substantially shorter than Yoccoz' proof.

*Fixed Points of Polynomial Maps*

I. Rotation Sets

II. Fixed Point Portraits

I. Rotation Sets

II. Fixed Point Portraits

I. We give a combinatorial analysis of rational rotation subsets of the circle. These are invariant subsets that have well-defined rational rotation numbers under the standard self-covering maps of $S^1$. This analysis has applications to the classification of dynamical systems generated by polynomials in one complex variable.

II. Douady, Hubbard and Branner have introduced the concept of a "limb" in the Mandelbrot set. A quadratic map $f(z)=z^2+c$ belongs to the $p/q$ limb if and only if there exist q external rays of its Julia set which land at a common fixed point of $f$, and which are permuted by $f$ with combinatorial rotation number $p/q$ in $Q/Z$, $p/q \neq 0$). (Compare Figure 1 and Appendix C, as well as Lemma 2.2.) This note will make a similar analysis of higher degree polynomials by introducing the concept of the "fixed point portrait" of a monic polynomial map.

*New Examples of Manifolds with Completely Integrable Geodesic Flows*

We construct Riemannian manifolds with completely integrable geodesic flows, in particular various nonhomogeneous examples. The methods employed are a modification of Thimm's method, Riemannian submersions and connected sums.

*Continuity of Convex Hull Boundaries*

In this paper we consider families of finitely generated Kleinian groups {$G_\mu$} that depend holomorphically on a parameter μ which varies in an arbitrary connected domain in $ \mathbb{C}$. The groups $G_\mu$ are quasiconformally conjugate. We denote the boundary of the convex hull of the limit set of $G_\mu$ by $\partial C(G_\mu)$. The quotient $\partial C(G_\mu)/G_\mu$ is a union of pleated surfaces each carrying a hyperbolic structure. We fix our attention on one component Sμ and we address the problem of how it varies with μ. We prove that both the hyperbolic structure and the bending measure of the pleating lamination of $S_\mu$ are continuous functions of $\mu$.