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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of $S-$unimodal maps is classified according to a distortion property, called the Markov-property.
The Milnor problem on one-dimensional attractors is solved for $S-$unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $\omega(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.
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.
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.
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 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.
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.
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.
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.
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.