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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.
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
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.
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.
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.