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.
This paper surveys various results concerning stability for the dynamics of Lagrangian (or Hamiltonian) systems on compact manifolds. The main, positive results state, roughly, that if the configuration manifold carries a hyperbolic metric, $\textit {i.e.}$ a metric of constant negative curvature, then the dynamics of the geodesic flow persists in the Euler-Lagrange flows of a large class of time-periodic Lagrangian systems. This class contains all time-periodic mechanical systems on such manifolds. Many of the results on Lagrangian systems also hold for twist maps on the cotangent bundle of hyperbolic manifolds. We also present a new stability result for autonomous Lagrangian systems on the two torus which shows, among other things, that there are minimizers of all rotation directions. However, in contrast to the previously known $\textit{hedlund}$ case of just a metric, the result allows the possibility of gaps in the speed spectrum of minimizers. Our negative result is an example of an autonomous mechanical Lagrangian system on the two-torus in which this gap actually occurs. The same system also gives us an example of a Euler-Lagrange minimizer which is not a Jacobi minimizer on its energy level.
This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler-Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry-Mather theory of twist maps and the Hedlund-Morse-Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.
We extend Sullivan's complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local connectivity of the corresponding Julia sets follows.
This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following
Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with a-priori bounds.
As a corollary, such maps are combinatorially and topologically rigid, and as a consequence, the Mandelbrot set is locally connected at the corresponding parameter values.
We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point, i.e., the combinatorial description of the boundary of chaos coincides with the topological description. We also show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable.
We use the methods developed with M. Lyubich for proving complex bounds for real quadratics to extend E. De Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows. In the Appendix we give an application of the complex bounds for proving local connectivity of some Julia sets.
Varchenko conjectured that, under certain genericity conditions, the number of critical points of a product $\phi$ of powers of linear functions on $\mathbb {C}^n$ should be given by the Euler characteristic of the complement of the divisor of $\phi$ (i.e., a union of hyperplanes). In this note two independent proofs are given of a direct generalization of Varchenko's conjecture to the case of a generalized meromorphic function on an algebraic manifold whose divisor can be any (generally singular) hypersurface. The first proof uses characteristic classes and a formula of Gauss--Bonnet type for affine algebraic varieties. The second proof uses Morse theory.
Let $M$ be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let $\Lambda(M)$ be the supremum of $\lambda_0(N)$ where $N$ varies over all hyperbolic 3-manifolds homeomorphic to the interior of $N$. Similarly, we let $D(M)$ be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of $M$. We observe that $\Lambda(M)=D(M)(2-D(M))$ if $M$ is not handlebody or a thickened torus. We characterize exactly when $\Lambda(M)=1$ and $D(M)=1$ in terms of the characteristic submanifold of the incompressible core of $M$.
Consider a planar Brownian motion run for finite time. The frontier or "outer boundary" of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension $4/3$, but this is still open). The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling of the plane by fractal tiles known as Gosper Islands.
We study the the tangent family $\mathcal{F} = \{\lambda \tan z, \lambda \in \mathbb{C} - \{0\}\}$ and give a complete classification of their stable behavior. We also characterize the the hyperbolic components and give a combinatorial description their deployment in the parameter plane.