Showing papers in "Acta Mathematica in 1999"

Partial hyperbolicity and robust transitivity

Abstract: Throughout this paper M denotes a three-dimensional boundaryless compact manifold and Diff(M) the space of gl-diffeomorphisms defined on M endowed with the usual Cl-topology. A ~-invariant set A is transitive if A=w(x) for some xEA. Here w(x) is the forward limit set of x (the accumulation points of the positive orbit of x). The maximal invariant set of ~ in an open set U, denoted by A~(U), is the set of points whose whole orbit is contained in U, i.e. A ~ ( U ) = ~ i e z ~i(U). The set A~(U) is robustly transitive if Ar is transitive for every diffeomorphism r CLclose to ~. A diffeomorphism ~EDiff(M) is transitive if M=w(x) for some xEM, i.e. if A ~ ( M ) = M is transitive. Analogously, ~ is robustly transitive if every r gLclose to also is transitive, i.e. if A ~ ( M ) = M is robustly transitive. In this paper we focus our attention on forms of hyperbolicity (uniform, partial and strong partial) of a maximal invariant set A~(U) derived from its robust transitivity. Observe that U can be equal to M, and then ~ is robustly transitive. On one hand, in dimension one there do not exist robustly transitive diffeomorphisms: the diffeomorphisms with finitely many hyperbolic periodic points (Morse~ Smale) are open and dense in Diff(S1). On the other hand, for two-dimensional diffeomorphisms, every robustly transitive set A~(U) is a basic set (i.e. A~(U) is hyperbolic, transitive, and the periodic points of ~ are dense in A~(U)). In particular, every robustly transitive surface diffeomorphism is Anosov and the unique surface which supports such diffeomorphisms is the torus T 2. These assertions follow from [M3] and [M4]. In dimension bigger than or equal to three, besides Anosov (hyperbolic) diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type. As far as we know, three types of such diffeomorphisms have been constructed: skew products,

Power-law subordinacy and singular spectra I. Half-line operators

Abstract: We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrodinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19].

On the structure of the Selberg class, I: 0≤d≤1

Abstract: The Selberg class S is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global L-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of S. Such a classication is based on a real-valued invariant d called degree, and the degree conjecture asserts that d 2 N for every L-function in S. The degree conjecture has been proved for d < 5=3, and in this paper we extend its validity to d < 2. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the L-functions in S.

On the dirichlet problem for degenerate Monge-Ampère equations

Abstract: We prove global second derivative estimates for the Dirichlet problem for degenerate Monge-Ampere equations which yield corresponding existence and regularity results. Our conditions are essentially optimal and our techniques, while drawing on previous investigations, are substantially new.

The deformation theorem for flat chains

Abstract: We prove that the deformation procedure of Federer and Fleming gives good approximations to arbritrary flat chains, not just those of finite mass and boundary mass. This implies, for arbitrary coefficient groups, that flat chains of finite mass and finite size are rectifiable, and also, for finite coefficient groups, that flat chains supported in sets of finite Hausdorff measure (or even finite integral geometric measure) have finite mass. The deformation theorem of Federer and Fleming [FF] is a fundamental tool in geometric measure theory. The theorem gives a way of approximating (in the so-called flat norm) a very general k-dimensional surface (flat chain) A in R by a polyhedral surface P consisting of k-cubes from a cubical lattice in R . Unfortunately, the theorem requires the original surface to have finite mass and finite boundary mass. In this paper, we remove these finiteness restrictions. That is, we show (in §1.1, 1.2, and 1.3) that the Federer-Fleming deformation procedure gives good approximations to an arbitrary flat chain A. Also, the approximating polyhedral surface depends only on the way in which typical translates of A intersect the (N − k) skeleton of the dual lattice. This lets us answer several open questions about flat chains: (1) For an arbitrary coefficient group, a nonzero flat k-chain cannot be supported in a set of k-dimensional measure 0. (2) For an arbitrary coefficient group, a flat chain of finite mass and finite size must be rectifiable. In particular, for any discrete group, finite mass implies rectifiability. (3) Let G be a normed group with sup{|g| : g ∈ G} = λ < ∞. Then for any flat chain with coefficients in G, M(A) ≤ λH(sptA). (Special cases of (1) and (2) are mentioned as open questions in [FL], and the special case G = Zp of (3) is mentioned as an open question in [F1]. Federer and Fleming [FF] proved (1) for real flat chains (and therefore also for integral flat chains). Almgren [A] introduced the notion of size and proved (2) for real flat chains; Federer [F2] then gave a much shorter proof.) 1991 Mathematics Subject Classification. Primary 49Q15; secondary 49Q20. The author was partially funded by NSF grants DMS-95-04456 and DMS-98-03493.

Quantum hyperplane section theorem for homogeneous spaces

Abstract: Author(s): Kim, Bumsig | Abstract: We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete intersections in homogeneous manifolds. For the proof we followed the scheme of Givental's proof of a mirror theorem for toric complete intersections.

The zero multiplicity of linear recurrence sequences

Abstract: Wri te k p(z) = II(z0 (1.2) i=1 with d i s t inc t roo ts c~1, ..., ak . The sequence is said to be nondegenerate if no quot ien t ~i/c~j ( l ~ i < j < ~ k ) is a root of 1. The zero multiplicity of the sequence is the number of n E Z wi th Un=O. For an i n t roduc t ion to l inear recurrences and exponen t i a l equat ions , see [10]. A classical t heo rem of Skolem Mahle r Lech [4] says t h a t a nondegene ra t e l inear recurrence sequence has finite zero mul t ip l ic i ty . Schlickewei [6] and van der Poo r t e n and Schlickewei [5] der ived uppe r bounds for the zero mul t ip l i c i ty when the me mbe r s of the sequence lie in a number field K . These bounds d e p e n d e d on the order t, the degree of K , as well as on the number of d i s t inc t p r ime ideal factors in the decompos i t i on of the f rac t ional ideals (c~) in K . More recently, Schlickewei [7] gave bounds which depend

Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces

Abstract: We prove that a germ of a holomorphic map f between C n and C n ' sending one real-algebraic submanifold MC n into another M ' � C n ' is algebraic provided Mcontains no complex-analytic discs and M is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.

Fourier inversion on a reductive symmetric space

Abstract: Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we establish an inversion formula for this transform.