*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.

*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.

*Scalings in Circle Maps (I)*

Let f be a flat spot circle map with irrational rotation number. Located at the edges of the flat spot are non-flat critical points $(S: x \rightarrow Ax^v ,v \geq 1)$. First, we define scalings associated with the closest returns of the orbit of the critical point. Under the assumption that these scalings go to zero, we prove that the derivative of long iterates of the critical value can be expressed in the scalings. The asymptotic behavior of the derivatives and the scalings can then be calculated. We concentrate on the cases for which one can prove the above assumption. In particular, let one of the singularities be linear. These maps arise for example as the lower bound of the non-decreasing truncations of non-invertible bimodal circle maps. It follows that the derivatives grow at a sub-exponential rate.

*Dynamics in One Complex Variable: Introductory Lectures*

These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 1989-90. These lectures are intended to introduce the reader to some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry.

*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.

*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.

*Shadowing Property for Nondegenerate Zero Entropy Piecewise Monotone Maps*

Let *f* be a continuous piecewise monotone map of the interval. If any two periodic orbits of *f* have different itineraries with respect to the partition of the turning points of *f*, then *f* is referred to as "nondegenerate". In this paper we prove that a nondegenerate zero entropy continuous piecewise monotone map *f* has the Shadowing Property if and only if 1) *f*dows not have neutral periodic points; 2) for each turning point *c* of *f*, either the ω-limit set ω(*c*,*f*) of *c* contains no periodic repellors or every periodic repellor in ω(*c*,*f*) is a turning point of *f* in the orbit of *c*. As an application of this result, the Shadowing Property for the Feigenbaum map is proven.

*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 $\mu$.

*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.

*One-Dimensional Maps and Poincare Metric*

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps. (AMS subject code 26A18)