Welcome to the preprint server of the Institute for Mathematical Sciences at Stony Brook University.
The IMS preprints are also available from the mathematics section of the arXiv e-print server, which offers them in several additional formats. To find the IMS preprints at the arXiv, search for Stony Brook IMS in the report name.
PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent greater than 1, has periodic points of any combinatorial type.
We prove geometric and scaling results for the real Fibonacci parameter value in the quadratic family $f_c(z) = z^2+c$. The principal nest of the Yoccoz parapuzzle pieces has rescaled asymptotic geometry equal to the filled-in Julia set of $z^2-1$. The modulus of two such successive parapuzzle pieces increases at a linear rate. Finally, we prove a "hairiness" theorem for the Mandelbrot set at the Fibonacci point when rescaling at this rate.
This is a continuation of notes on dynamics of quadratic polynomials. In this part we transfer the our prior geometric result to the parameter plane. To any parameter value c in the Mandelbrot set (which lies outside of the main cardioid and little Mandelbrot sets attached to it) we associate a "principal nest of parapuzzle pieces" and show that the moduli of the annuli grow at least linearly. The main motivation for this work was to prove the following:
Theorem B (joint with Martens and Nowicki): Lebesgue almost every real quadratic polynomial which is non-hyperbolic and at most finitely renormalizable has a finite absolutely continuous invariant measure.
A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.
We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points.
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.