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.
We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex numbers ${\textbf C}$ to itself which have degree two or more in each copy. As a consequence of these results we prove a transitivity relation between hyperbolic components in parameter space which was conjectured by Milnor.
We study the Teichmüller metric on the Teichmüller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichmüller metric is approximated up to bounded additive distortion by the sup metric on a product of lower dimensional spaces. The main technical tool in the proof is the use of estimates of extremal lengths of curves in a surface based on the geometry of their hyperbolic geodesic representatives.
We introduce two models, the Fermi-Ulam model in an external field and a one dimensional system of bouncing balls in an external field above a periodically oscillating plate. For both models we investigate the possibility of unbounded motion. In a special case the two models are equivalent.
In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve G in the plane. It is shown that there do not exist invariant circles near G when there is a point on G where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not $C^1$ there are examples with orbits that converge to a point of G. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards.
A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi$. The main theorem we will prove is
Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has "many" periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $\textbf{S}^1$.
We will also prove some refinements of Theorem 1: the "well distribution" of periodic orbits in the polygon and the residuality of the points $q \in Q$ with a dense set of periodic directions.
We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only if the geodesic is precompact, i.e. its image in the moduli space is contained in a compact set. Some applications are given, e.g. to stability properties of certain quasi-geodesics in Teichmüller space, and to estimates of translation distance for pseudo-Anosov maps.
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
In $ \mathit {Ch91a}$ it was shown that the billiard ball map for the periodic Lorentz gas has infinite topological entropy. In this article we study the set of points with infinite Lyapunov exponents. Using the cell structure developed in $ \mathit {BSC90,Ku}$ we construct an ergodic invariant probability measure with infinite topological entropy supported on this set. Since the topological entropy is infinite this is a measure of maximal entropy. From the construction it is clear that there many such measures can coexist on a single component of topological transitivity. We also construct an ergodic invariant probability measure with finite entropy which is supported on this set showing that infinite exponents do not necessarily lead to infinite entropy.
For the polynomials $p_c(z)=z^d+c$, the periodic points of periods dividing $n$ are the roots of the polynomials $P_n(z)=p_c^{\circ n}(z)-z$, where any degree $d\geq 2$ is fixed. We prove that all periodic points of any exact period $k$ are roots of the same irreducible factor of $P_n$ over $\mathbb{C}(c)$. Moreover, we calculate the Galois groups of these irreducible factors and show that they consist of all permutations of periodic points which commute with the dynamics. These results carry over to larger families of maps, including the spaces of general degree-$d$-polynomials and families of rational maps. Main tool, and second main result, is a combinatorial description of the structure of the Mandelbrot set and its degree-$d$-counterparts in terms of internal addresses of hyperbolic components. Internal addresses interpret kneading sequences of angles in a geometric way and answer Devaney's question: "How can you tell where in the Mandelbrot a given rational external ray lands, without having Adrien Douady at your side?"