Showing papers in "Annals of Mathematics in 1998"

Statistical properties of dynamical systems with some hyperbolicity

Abstract: This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970’s (see [S1], [B], [R2]). Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY1]). Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents. The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps.

Handlebody construction of Stein surfaces

Abstract: The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained-they correspond to open handlebodies with all handles of index < 2. An uncountable collection of exotic l4,'S is shown to admit Stein structures. New invariants of contact 3-manifolds are produced, including a complete (and computable) set of invariants for determining the homotopy class of a 2-plane field on a 3-manifold. These invariants are applicable to Seiberg-Witten theory. Several families of oriented 3-manifolds are examined, namely the Seifert fibered spaces and all surgeries on various links in S3, and in each case it is seen that "most" members of the family are the

On decay of correlations in Anosov flows

Abstract: There is some disagreement about the meaning of the phrase 'chaotic flow.' However, there is no doubt that mixing Anosov flows provides an example of such systems. Anosov systems were introduced and extensively studied in his classical memoir ([A]). Among other things he proved the following fact known now as Anosov alternative for flows: Either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an Anosov diffeomorphism by a constant roof function. If the first alternative holds gt is mixing with respect to every Gibbs measure (see [PP2]). Now the natural question is: What is the estimated rate of mixing? This is certainly one of the simplest questions concerning correlation decay in continuous time systems. Nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure. The easier case of Anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of Sinai, Ruelle and Bowen ([B2]). Namely, one uses Markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing. This method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations. However, the simplest example-suspensions with locally constant roof functions-never have such a property ([RI]). One can use the above observation to produce examples of Axiom A flows with arbitrary slow correlation decay. It became clear therefore that some additional geometric properties should be taken into account. In recent work Chernov ([Chl], [Ch2]) has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature. His method relies on the technique of Markov approximation developed in [Chl]. The aim of this paper is to combine geometric considerations of Chernov with the thermodynamic formalism approach. The later method seems to be more appropriate than Markov approximations since it gives simpler

Quasi-periodic solutions of hamiltonian perturbations of 2d linear schrodinger equations

Abstract: The general problem discussed here is the persistency of quasi-periodic solutions of linear or integrable equations after Hamiltonian perturbation. This subject is closely related to the well-known "KAM-theory" of invariant tori in smooth dynamical systems. The main source of new difficulties is the fact that one considers here (finite dimensional) tori in an infinite dimensional phase space. For instance, already in the study of time periodic solutions, the fact that the phase space is infinite dimensional leads to small divisor problems. Now the standard KAM-technology may be reworked in a form applicable to treat persistency problems in space dimension (1D) in the case of Dirichlet boundary conditions say, where in particular the normal frequencies of the problem are well-separated. Results along this line were obtained by S. Kuksin [Ki] and C. E. Wayne [W] among other works. We mentioned [Ki] as the main expository reference for the KAM approach. Considering PDE's in iD with periodic boundary conditions and especially in space dimension > 1, a significant new problem arises due to the presence of clusters of normal frequencies.

Flows on homogeneous spaces and Diophantine approximation on manifolds

Abstract: We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.

The Novikov conjecture for groups with finite asymptotic dimension

Abstract: Recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology (page 28, [14]). More precisely, the asymptotic dimension for a metric space is the smallest integer n such that for any r > 0, there exists a uniformly bounded cover C = {Ui}iEI of the metric space for which the r-multiplicity of C is at most n + 1; i.e., no ball of radius r in the metric space intersects more than n + 1 members of C [14]. The class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word-length metric, then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word-length metrics (cf. Section 6). This, together with a result of Gromov in [14], implies that finitely generated subgroups of Gromov's hyperbolic groups have finite asymptotic dimension. Currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known. It should also be noted that two different definitions of asymptotic dimension

The dynamics on three-dimensional strictly convex energy surfaces

Abstract: We show that a Hamiltonian flow on a three-dimensional strictly convex energy surface S C R4 possesses a global surface of section of disc type. It follows, in particular, that the number of its periodic orbits is either 2 or oc, by a recent result of J. Franks on area-preserving homeomorphisms of an open annulus in the plane. The construction of this surface of section is based on partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into R x S3 equipped with special almost complex structures.

The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets

Abstract: It is shown that the boundary of the Mandelbrot set M has Hausdorff dimension two and that for a generic c E AM, the Julia set of z I > Z2 + C also has Hausdorff dimension two. The proof is based on the study of the bifurcation of parabolic periodic points.

Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation

Abstract: The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is "no" in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak [7] suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: "If the level set topology in the temperature field for the 2D quasi-geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle, then no finite time singularity is possible." Numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle. There is a naturally associated notion of simple hyperbolic saddle breakdown. The main theorem we present in this paper shows that such breakdown cannot occur in finite time. We also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time. Using the same techniques, we see that analogous results hold for incompressible 2D and 3D Euler. These results were announced in [9], but with a slight difference in the definition of a simple hyperbolic saddle. The definition given in Section 4 generalizes the one given in the announcement. See also Constantin [4], discussed in Section 7, Remark 5 below. The whole work described in this paper is basically part of the author's thesis. I am particularly grateful to my thesis advisor Charles Fefferman for his attention, support, guidance and advice. I am indebted to D. Christodoulou and P. Constantin for helpful corrections and suggestions. I wish to thank A. Majda for suggesting the subject and E. Tabak for discussions and com-

Positivite et discretion des points algebriques des courbes

Abstract: We prove the discreteness of algebraic points (with respect to the Neron-Tate height) on a curve of genus greater than one embedded in his jacobian. This result was conjectured by Bogomolov. We also prove the positivity of the self intersection of the admissible dualizing sheaf.

On Culler-Shalen seminorms and Dehn filling

Abstract: If F is a finitely generated discrete group and G a complex algebraic Lie group, the G-character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in G. Marc Culler and Peter Shalen developed the theory of SL2(C)-character varieties of finitely generated groups and applied their results to study the topology of 3-dimensional manifolds in the papers [6], [7], [8]. Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. The Mostow rigidity theorem implies that the holonomy representation p: iri(M) Isom+(H3) _ PSL2(C) is unique up to conjugation and taking complex conjugates. The orientability of M can be used to show 1 lifts to a representation p E Hom(-ri (M), SL2(C)) whose character determines an essentially unique point of Xp of X(iri(M)), the SL2(C)-character variety of irl(M). Culler and Shalen [8] proved that the component X0 of X(X1ri(M)) which contains X' is a curve. One of their major contributions was to show how X0 determines a norm on H1(WM; R) which encodes many topological properties of M. In particular it provides information on the Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL2(C)-character variety of a connected, compact, orientable, irreducible 3manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a seminorm. The first half of this paper is devoted to the development of the general theory of -Culler-Shalen seminorms defined for curves of PSL2(C)-characters. By working over PSL2(C) we obtain a theory that is more generally applicable than its SL2(C) counterpart, while being only mildly more difficult to set up. In the second half of this paper we apply the theory of Culler-Shalen seminorms to study the Dehn filling operation. In particular we examine the relationship between fillings which yield manifolds having a positive dimen-

Matching rules and substitution tilings

Abstract: A substitution tiling is a certain globally de ned hierarchical structure in a geometric space; we show that for any substitution tiling in En, n > 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, in nite collections of forced aperiodic tilings are constructed. Figure 1: A substitution tiling On the left in gure 1, L-shaped tiles are repeatedly \in ated and subdivided". (We de ne our terms more precisely in Section 1.) As this process is iterated, larger and larger regions of the plane are tiled with L-tiles hierarchically Dept. of Mathematics, Univ. Arkansas, Fayetteville, AR 72701. This work was partially supported by the Geometry Center under NSF Grant DMS-8920161.

The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds

Abstract: In this paper we prove a conjecture of A. Katok, stating that the geodesic flow on a compact rank 1 manifold admits a uniquely determined invariant measure of maximal entropy. This generalizes previous work of R. Bowen and G. Margulis. As an application we show that the exponential growth rate of the set of singular closed geodesics is strictly smaller than the topological entropy.

Markov approximations and decay of correlations for Anosov flows

Abstract: We develop Markov approximations for very general suspension flows. Based on this, we obtain a stretched exponential bound on time correlation functions for 3-D Anosov flows that verify ‘uniform nonintegrability of foliations’. These include contact Anosov flows and geodesic flows on compact surfaces of variable negative curvature. Our bound on correlations is stable under small smooth perturbations.

Theorie d’Iwasawa des représentations de de Rham d’un corps local

Abstract: Sleep inducers of the formula: wherein R is hydrogen or halo of atomic weight of from 18 to 36, R' is hydrogen, halo or CF3 and R DEG is hydrogen or lower alkyl. Preparation by cyclizing a 2-nitro-phenylacetic acid is also disclosed.

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface

Abstract: Let n and d be coprime positive integers, and define M(n,d) to be the moduli space of (semi)stable holomorphic vector bundles of rank n, degree d and fixed determinant on a compact Riemann surface E. This moduli space is a compact Kifhler manifold which has been studied from many different points of view for more than three decades (see for instance Narasimhan and Seshadri [41]). The subject of this article is the characterization of the intersection pairings in the cohomology ring1 H* (M (n, d)). A set of generators of this ring was described by Atiyah and Bott in their seminal 1982 paper [2] on the YangMills equations on Riemann surfaces (where in addition inductive formulas for the Betti numbers of M (n, d) obtained earlier using number-theoretic methods [13], [25] were rederived). By Poincare duality, knowledge of the intersection pairings between products of these generators (or equivalently knowledge of the evaluation on the fundamental class of products of the generators) completely determines the structure of the cohomology ring. In 1991 Donaldson [15] and Thaddeus [47] gave formulas for the intersection pairings between products of these generators in H* (M (2, 1)) (in terms of Bernoulli numbers). Then using physical methods, Witten [50] found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in H* (M (n, d)) for general rank n. These generalized his (rigorously proved) formulas [49] for the symplectic volume of M (n, d): For instance, the symplectic volume of M (2, 1) is given by

Formule de Plancherel pour les espaces symétriques réductifs

Abstract: I1 nous semble utile de donner le plan d'ensemble de notre demonstration de la formule de Plancherel et de la situer par rapport a d'autres travaux. Nous avons d'abord, en collaboration avec J. L. Brylinski [BD] (grace aux D-modules) puis avec J. Carmona [CD1] (grace aux foncteurs de translation et a la these de Bruhat), etabli le prolongement meromorphe des integrales d'Eisenstein. Dans le cas des sous-groupes paraboliques UO-stables minimaux, ces resultats sont dus a G. Olafsson [01] et E. van den Ban [B1], [B2]. I1 faut aussi mentionner le travail de T. Oshima et J. Sekiguchi [OS] sur les espaces de type G/IK. L'etape suivante a ete la preuve de la temperance et de majorations, sur l'axe imaginaire pur, du prolongement meromorphe des integrales d'Eisenstein [D1] (cf. [B2] pour le cas des sous-groupes paraboliques a9-stables minimaux). Pour la temperance, l'utilisation de la theorie des series discretes de T. Oshima et T. Matsuki [OM], initiee par M. Flensted-Jensen et basee sur l'importante dualite introduite par celui-ci [FJ], joue un role determinant. I1 en va de meme d'un critere de temperance de T. Oshima [0] et de l'utilisation des foncteurs de translation dans diff6rentes situations (voir [V]). Les majorations sont deduites de la temperance en utilisant une technique d'E. van den Ban [B2], qui s'adapte facilement a notre situation. Dans [BCD], en collaboration avec E. van den Ban et J. Carmona, nous avons introduit la notion de fonction IIhol (pour rappeler une terminologie de Harish-Chandra), forme les paquets d'ondes dans l'espace de Schwartz et obtenu un critere pour qu'une fonction soit IIHh'. Les integrales d'Eisenstein, multipliees par un polynome convenable, sont des fonctions IIhl. Cette partie est tres proche d'une partie du travail [H-Cl] de Harish-Chandra pour le cas des groupes. Ce developpement a ete rendu possible par la theorie du terme constant de J. Carmona [C].

The polynomial $X^2+Y^4$ captures its primes

Abstract: This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} \Lambda(a^2 + b^4) = 4\pi^{-1}\kappa x^{3/4} (1 + O(\log\log x / \log x)) where a, b run over positive integers and \kappa = \int^1_0 (1 - t^4)^{1/2} dt = \Gamma(1/4)^2 /6\sqrt{2\pi}. Here of course, \Lambda denotes the von Mangoldt function and \Gamma the Euler gamma function.

Discrete decomposability of the restriction of A_q(λ)with respect to reductive subgroups II-micro-local analysis and asymptotic K-support

Abstract: Let G' c G be real reductive Lie groups. This paper offers a criterion on the triplet (G, G', ir) that the irreducible unitary representation ir of G splits into a discrete sum of irreducible unitary representations of a subgroup G' when restricted to G', each of finite multiplicity. Furthermore, we shall give an upper estimate of the multiplicity of an irreducible unitary representation of G' occurring in WrIG'

Local connectivity of the Julia set of real polynomials

Abstract: One of the main questions in the field of complex dynamics is the question whether the Mandelbrot set is locally connected, and related to this, for which maps the Julia set is locally connected. In this paper we shall prove the following Main Theorem: Let $f$ be a polynomial of the form $f(z)=z^d +c$ with $d$ an even integer and $c$ real. Then the Julia set of $f$ is either totally disconnected or locally connected. In particular, the Julia set of $z^2+c$ is locally connected if $c \in [-2,1/4]$ and totally disconnected otherwise.

Uniform estimates on the number of collisions in semi-dispersing billiards

Abstract: Estimating the number of collisions in a neighborhood of a given point is a central problem in the theory of billiard systems. Moreover, the existence of a uniform estimate on the number of collisions is related to various important properties of a billiard system. For example, the Sinai-Chernov formulas for the metric entropy of billiards are proved under the assumption that such an estimate exists ([3], [9]). It is quite obvious that if the boundary of the billiard has any concave parts then a trajectory can have arbitrarily many collisions in a neighborhood of a "point of concavity," and sometimes even infinitely many collisions (see, for example [11]). Therefore, the class of billiards for which uniform estimates are possible are the semi-dispersing billiards, which have been studied extensively in the numerous works (see the review in [7]). The most important examples of semi-dispersing billiards are the billiards that correspond to hard balls gas models. Vaserstein (1979, [11]) and Gal'perin (1981, [4]) proved that in semidispersing billiards any trajectory has only finitely many collisions in any finite period of time. Sinai (1978, [8]) proved the existence of a uniform estimate for billiards inside polyhedral angles, and pointed out that his results should also hold for semi-dispersing billiards in a neighborhood of a point x with linearly independent normals to the "walls" of the billiard at x (see the remark at the end of [8]). In this paper we deal with semi-dispersing billiards on arbitrary manifolds. All our results apply to the usual billiards in Rk and Wk. First we prove that for semi-dispersing billiards on arbitrary manifolds any trajectory has only a finite number of collisions in a finite period of time.

On the classification of unitary representations of reductive Lie groups

Abstract: Each subset is identified conjecturally (Conjecture 0.6) with a collection of unitary representations of a certain subgroup G(λu) of G. (We will give strong evidence and partial results for this conjecture.) In this way the problem of classifying Πu(G) would be reduced (by induction on the dimension of G) to the case G(λu) = G. Before considering the general program in more detail, we describe it in the familiar case G = SL(2,R). (This example will be treated more com-

Fourier coefficients of half-integral weight modular forms modulo $\ell$

Abstract: For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes $\ell$ there are infinitely many square-free integers $m$ for which $|c(m)|_\ell = 1$. Consequently we obtain indivisibility results for ``algebraic parts'' of central critical values of modular $L$-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for $L$-function values. For example if $\Delta(z)$ is Ramanujan's cusp form and $g(z)=\sum_{n=1}^{\infty}c(n)q^n$ is the cusp form for which $$L(\Delta_D,6)=\fracwithdelims(){\pi}{D}^6\frac{\sqrt{D}}{5!}\frac{ } { }\cdot c(D)^2,$$ for fundamental discriminants $D>0,$ then for $N\geq 1$ $$\sum_{k=-\infty}^\infty c(N-k^2) \equiv \half \sum_{d|N}(\chi_{-1}(d)+\chi_{-1}(N/d))d^6 \pmod {61}. \tag{0}$$

The periodic points of renormalization

Abstract: It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent a > 1, has periodic points of any combinatorial type. A central question in the theory of dynamical systems is whether small scale geometrical properties of dynamical systems are determined by the combinatorial properties of the system. Indeed, such universality of small scale geometry was discovered by Coullet-Tresser and Feigenbaum. They studied infinitely renormalizable unimodal maps of the period doubling type and observed that the geometry of the invariant Cantor set of such maps converges when looking at smaller and smaller scales. Furthermore they observed that the limiting geometry was universal, in the sense that the small scale geometry of these Cantor sets depends only on the local behavior of the map around the critical point. This local behavior is specified by the critical exponent. To explain the universality of geometry, they defined the period doubling renormalization operator on a suitable space of unimodal maps. This operator acts like a microscope: the image under the renormalization operator is a unimodal map describing the geometry and dynamics on a smaller scale. The universality of geometry was understood by conjecturing that the renormalization operator has a unique hyperbolic fixed point. In particular, the infinitely renormalizable unimodal maps form the stable manifold of the fixed point of the renormalization operator. The first step in proving these conjectures is showing the existence of a renormalization fixed point.

Lattice gases, large deviations, and the incompressible Navier-Stokes equations

Abstract: We study the incompressible limit for a class of stochastic particle systems on the cubic lattice Zd, d = 3. For initial distributions corresponding to arbitrary macroscopic L2 initial data, the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible Navier-Stokes equations. Furthermore, explicit exponential rates for the convergence (large deviations) are obtained. The probability to violate the divergence-free condition decays at rate at least exp{-E-d+l}, while the probability to violate the momentum conservation equation decays at rate exp{fE-d+2} with an explicit rate function given by an HI1 norm.