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

* Starred papers have appeared in the journal cited.

*A Partial Description of the Parameter Space of Rational Maps of Degree Two: Part 2*

This continues the investigation of a combinatorial model for the variation of dynamics in the family of rational maps of degree two, by concentrating on those varieties in which one critical point is periodic. We prove some general results about nonrational critically finite degree two branched coverings, and finally identify the boundary of the rational maps in the combinatorial model, thus completing the proofs of results announced in Part 1.

*Polynomial Root-Finding Algorithms and Branched Covers*

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that computes an $\epsilon$-factorization of the polynomial which has an arithmetic complexity of $ \mathcal{O} (d^2(log d)^2 + d(log d)^2|log \epsilon |)$. At the present time (1993), this complexity is the best known in terms of the degree.

*Expanding Direction of the Period Doubling Operator*

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a "Perron-Frobenius type operator'", to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.

*The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets*

It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \partial M$, the Julia set of $z \mapsto z^2 + c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.

*Critical Circle Maps Near Bifurcation*

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Hölder continuous function of the parameter. AMS subject code: 54H20

*The Teichmuller Space of an Anosov Diffeomorphism of T*

^{2}.In this paper we consider the space of smooth conjugacy classes of an Anosov diffeomorphism of the two-torus. The only 2-manifold that supports an Anosov diffeomorphism is the 2-torus, and Franks and Manning showed that every such diffeomorphism is topologically conjugate to a linear example, and furthermore, the eigenvalues at periodic points are a complete smooth invariant. The question arises: what sets of eigenvalues occur as the Anosov diffeomorphism ranges over a topological conjugacy class? This question can be reformulated: what pairs of cohomology classes (one determined by the expanding eigenvalues, and one by the contracting eigenvalues) occur as the diffeomorphism ranges over a topological conjugacy class? The purpose of this paper is to answer this question: all pairs of Hölder reduced cohomology classes occur.

*On the Lebesgue Measure of the Julia Set of a Quadratic Polynomial*

The goal of this note is to prove the following theorem: Let $p_a(z)=z^2+a$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then the Lebesgue measure of the Julia set $J(p_a)$ is equal to zero. As part of the proof we discuss a property of the critical point to be *persistently recurrent*, and relate our results to corresponding ones for real one dimensional maps. In particular, we show that in the persistently recurrent case the restriction $p_a|\omega (0)$ is topologically minimal and has zero topological entropy. The Douady-Hubbard-Yoccoz rigidity theorem follows this result.

*Ergodic Theory for Smooth One-Dimensional Dynamical Systems*

In this paper we study measurable dynamics for the widest reasonable class of smooth one dimensional maps. Three principle decompositions are described in this class : decomposition of the global measure-theoretical attractor into primitive ones, ergodic decomposition and Hopf decomposition. For maps with negative Schwarzian derivative this was done in the series of papers [BL1-BL5], but the approach to the general smooth case must be different.

*Dynamics of Certain Smooth One-Dimensional Mappings III: Scaling Function Geometry*

We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the critical points of a geometrically finite one-dimensional mapping form a complete set of $C^1$-invariants within a topological conjugacy class.

*Dynamics of Certain Smooth One-Dimensional Mappings IV: Asymptotic Geometry of Cantor Sets*

We study hyperbolic mappings depending on a parameter $\varepsilon$. Each of them has an invariant Cantor set. As $\varepsilon$ tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function geometry of the invariant Cantor set as $\varepsilon$ goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed $\sqrt{\varepsilon}$. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping is not expanding. Removing these discontinuities by continuous extension, we show that we obtain the scaling function of the limiting mapping with respect to the Ulam-von Neumann type metric.