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

* Starred papers have appeared in the journal cited.

*Scalings in Circle Maps II*

In this paper we consider one parameter families of circle maps with nonlinear flat spot singularities. Such circle maps were studied in [Circles I] where in particular we studied the geometry of closest returns to the critical interval for irrational rotation numbers of constant type. In this paper we apply those results to obtain exact relations between scalings in the parameter space to dynamical scalings near parameter values where the rotation number is the golden mean. Then results on [Circles I] can be used to compute the scalings in the parameter space. As far as we are aware, this constitutes the first case in which parameter scalings can be rigorously computed in the presence of highly nonlinear (and nonhyperbolic) dynamics.

*The Julia Sets and Complex Singularities in Hierarchical Ising Models*

We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known that the singularities of free energy of this model lie on the Julia set of some rational endomorphism $f$ related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin of $f$. We prove that for almost all (with respect to the harmonic measure) geodesics the complex critical exponent is common, and compute it.

*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 $\mu$.

*Dynamics of Certain Smooth One-Dimensional Mappings I: The $C^{1+\alpha }$-Denjoy-Koebe Distortion Lemma*

We prove a technical lemma, the $C^{1+\alpha }$-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a $C^{1+\alpha }$ one-dimensional mapping $f:M \rightarrowtail M$ with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.

*Dynamics of Certain Smooth One-Dimensional Mappings II: Geometrically Finite One-Dimensional Mappings*

We study geometrically finite one-dimensional mappings. These are a subspace of $C^{1+\alpha }$ one-dimensional mappings with finitely many, critically finite critical points. We study some geometric properties of a mapping in this subspace. We prove that this subspace is closed under quasisymmetrical conjugacy. We also prove that if two mappings in this subspace are topologically conjugate, they are then quasisymmetrically conjugate. We show some examples of geometrically finite one-dimensional mappings.

*Pleating Coordinates for the Maskit Embedding of the Teichmüaut;ller Space of Punctured Tori*

The authors introduce a new set of global parameters, which they call pleating coordinates, for the Teichmüller space $T_{1,1}$ of the punctured torus. The space $T_{1,1}$ is embedded as a holomorphic family $G_\mu$ of Kleinian groups, where the complex parameter μ varies in a simply connected domain M in the complex plane. This is done in such a way that the regular set $\Omega (G_\mu )$ has a unique invariant component $\Omega _0(G_\mu )$ and the points in $T_{1,1}$ are represented by the Riemann surface $\Omega (G_\mu)/G_\mu$. This embedding is in fact the Maskit embedding. The new coordinates are geometric in the sense that they are related to the geometry of the hyperbolic manifold $H^3/G_\mu$. More precisely, they can be read off from the geometry of the punctured torus $\partial C_0/G_\mu$, where $\partial C_0$ is the component of the convex hull boundary facing $\Omega _0(G_\mu)$. The surface $\partial C_0$ has a natural hyperbolic metric and is pleated along geodesics that project to a geodesic lamination $\lambda$ on $\partial C_0/G_\mu$. There is some specific choice of transverse measure for the pleating lamination $\lambda$, which allows the authors to introduce a notion of pleating length for $G_\mu$. The laminations and their pleating lengths are the coordinates for $M$.

*The Classification of Critically Preperiodic Polynomials as Dynamical Systems*

The object of this paper is to classify all polynomials $p$ with the properties that all critical points of $p$ are strictly preperiodic under iteration of $p$. We will also characterize the Julia sets of such polynomials.