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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
We show that for any unimodal polynomial $f$ with real coefficients, all conformal measures for $f$ are ergodic.
A S. N. Bernstein problem is solved under a natural irreducibility condition. Earlier this result was obtained only in some special case.
We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. These products were observed by J. Milnor in computer experiments which inspired Lavaurs' proof of non local-connectivity for the cubic connectedness locus. Cubic polynomials in such a product may be renormalized to produce a pair of quadratic maps. The inverse construction is an $\textit {intertwining surgery}$ on two quadratics. The idea of intertwining first appeared in a collection of problems edited by Bielefeld. Using quasiconformal surgery techniques of Branner and Douady, we show that any two quadratics may be intertwined to obtain a cubic polynomial. The proof of continuity in our two-parameter setting requires further considerations involving ray combinatorics and a pullback argument.
The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly. It is not hard to see that the complex is finite-dimensional, but locally infinite. It was introduced by Harvey as an analogy, in the context of Teichmüller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces. In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov.
In a certain sense this hyperbolicity is an explanation of why the Teichmüller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmüller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short. The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmüller space is "relatively hyperbolic" with respect to this family. A similar relative hyperbolicity result is proved for the mapping class group of a surface.
(revised version of January 1998)
The existence of smooth families of Lorenz maps exhibiting all possible dynamical behavior is established and the structure of the parameter space of these families is described.
In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.
We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on $T^2$ which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
We discuss some properties of a class of cellular automata sometimes called a "generalized ant". This system is perhaps most easily understood by thinking of an ant which moves about a lattice in the plane. At each vertex (or "cell"), the ant turns right or left, depending on the the state of the cell, and then changes the state of the cell according to certain prescribed rule strings. (This system has been the subject of several Mathematical Entertainments columns in the Mathematical Intelligencer; this article will be a future such column). At various times, the distributions of the states of the cells for certain ants is bilaterally symmetric; we categorize a class of ants for which this is the case and give a proof using Truchet tiles.
It is shown that a polynomial with a Cremer periodic point has a non-accessible critical point in its Julia set provided that the Cremer periodic point is approximated by small cycles.
Consider the group $Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\omega)$ with the Hofer $L^{\infty}$-norm. A path in $Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\mathcal{L}$. In this paper, we give a necessary condition for a path $\gamma$ to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for $\mathcal{L}$. This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky's work on the second variation formula. Using it, we construct a symplectomorphism of $S^2$ which cannot be reached from the identity by a shortest path. In later papers in this series, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We will also investigate conditions under which geodesics are absolutely length-minimizing.