Showing papers in "Israel Journal of Mathematics in 2006"

A local variational relation and applications

Abstract: In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infh μ(T,ℙ)≥h top(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andh μ(T,ℙ) > 0 for each ℙ finer thanu, then infh μ(T,ℙ > 0 andh top(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained. We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders).

Complexes of graph homomorphisms

Abstract: Hom(G, H) is a polyhedral complex defined for any two undirected graphsG andH. This construction was introduced by Lovasz to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes.

Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality

Abstract: In this paper we studynon-interactive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating.

Certain conjectures relating unipotent orbits to automorphic representations

Abstract: In this paper I formulate certain conjectures relating the structure of unipotent orbits to automorphic representations. We consider a few examples and prove some of these conjectures.

Forbidden paths and cycles in ordered graphs and matrices

Abstract: At most how many edges can an ordered graph ofn vertices have if it does not contain a fixed forbidden ordered subgraphH? It is not hard to give an asymptotically tight answer to this question, unlessH is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix)P. We disprove a general conjecture of Furedi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases whenP is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur amongn points in the plane isO(n 4/3).

There are significantly more nonegative polynomials than sums of squares

Abstract: We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of even powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two, it follows that there are significantly more nonnegative polynomials than sums of squares and there are significantly more sums of squares than sums of even powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases.

Implicit functions from topological vector spaces to Banach spaces

Abstract: We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.

Theory of valuations on manifolds, I. Linear spaces

Abstract: This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (non-translation invariant) valuations on a linear spaceV. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [4] obtained for polynomial valuations.

Prime arithmetic Teichmüller discs in H(2)

Abstract: It is well-known that Teichmuller discs that pass through “integer points” of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc.

Dynamical borel-cantelli lemma for hyperbolic spaces

Abstract: We prove that almost every (resp. almost no) geodesic rays in a finite volume hyperbolic manifold of real dimensionn intersects for arbitrary large timest a decreasing family of balls of radiusr t, provided the integral ∫ 0 ∞ r t n −1 dt diverges (resp. converges).

A logarithmic lower bound for multi-dimensional bohr radii

Abstract: We prove that the Bohr radiusKn of then-dimensional polydisc in ℂn is up to an absolute constant ≥ √logn/log logn/n. This improves a result of Boas and Khavinson.

Efficient calculation of Stark-Heegner points via overconvergent modular symbols

Abstract: This note presents a qualitative improvement to the algorithm presented in [DG] for computing Stark-Heegner points attached to an elliptic curve and a real quadratic field. This algorithm computes the Stark-Heegner point with ap-adic accuracy ofM significant digits in time which is polynomial inM, the primep being treated as a constant, rather than theO(p M ) operations required for the more naive approach taken in [DG]. The key to this improvement lies in the theory of overconvergent modular symbols developed in [PS1] and [PS2].

Gromov-Witten invariants and pseudo symplectic capacities

Abstract: We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.

Finite mean oscillation and the Beltrami equation

Abstract: We prove the existence and uniqueness of homeomorphic ACL solutions to the Beltrami equation in the case when the dilatation coefficient of the equation has a majorant of finite mean oscillation.

Branch rings, thinned rings, tree enveloping rings

Abstract: We develop the theory of “branch algebras”, which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field $$\Bbbk $$ % MathType!End!2!1! we contruct a $$\Bbbk - algebra$$ % MathType!End!2!1! which

Atomic surfaces, tilings and coincidence I. irreducible case

Abstract: An irreducible Pisot substitution defines a graph-directed iterated function system. The invariant sets of this iterated function system are called the atomic surfaces. In this paper, a new tiling of atomic surfaces, which contains Thurston’sβ-tiling as a subclass, is constructed. Related tiling and dynamical properties are studied. Based on the coincidence condition defined by Dekking [Dek], we introduce thesuper-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical properties of atomic surfaces. We conjecture that every Pisot substitution satisfies the super-coincidence condition.

Distance sets of well-distributed planar sets for polygonal norms

Abstract: LetX be a two-dimensional normed space, and letBX be the unit ball inX. We discuss the question of how large the set of extremal points ofBX may be ifX contains a well-distributed set whose distance set Δ satisfies the estimate |Δ∩[0,N]|≤CN3/2-e. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩[0,N]|≤CN.

Tire track geometry: variations on a theme

Abstract: We study closed smooth convex plane curves Λ enjoying the following property: a pair of pointsx, y can traverse Λ so that the distances betweenx andy along the curve and in the ambient plane do not change; such curves are calledbicycle curves. Motivation for this study comes from the problem how to determine the direction of the bicycle motion by the tire tracks of the bicycle wheels; bicycle curves arise in the (rare) situation when one cannot determine which way the bicycle went.

Minimal models for noninvertible and not uniquely ergodic systems

Abstract: Let (Y, S) be a (not necessarilly invertible) topological dynamical system on a zero-dimensional metric spaceY without periodic points. Then there exists a minimal system (X, T) with the same simplex of invariant measures as (Y, S). More precisely, there exists a Borel isomorphism between full sets inY andX such that the adjoint map on measures is a homeomorphism between the corresponding sets of invariant measures in the weak topology. As an application we construct a minimal system carrying isomorphic copies of all nonatomic invariant measures.

On certain multiplicity one theorems

Abstract: We prove several multiplicity one theorems in this paper. Fork a local field not of characteristic two, andV a symplectic space overk, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), whereV is an orthogonal space overk. Whenk is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).

Random complex zeroes, II. Perturbed lattice

Abstract: We show that the flat chaotic analytic zero points (i.e. zeroes of a random entire function\(\psi (z) = \sum {_{k = 0}^\infty \zeta } k\frac{{z^k }}{{\sqrt {k!} }}\) where ζ0, ζ1, … are independent standard complex-valued Gaussian variables) can be regarded as a random perturbation of a lattice in the plane. The distribution of the distances between the zeroes and the corresponding lattice points is shift-invariant and has a Gaussian-type decay of the tails.

The subgaussian constant and concentration inequalities

Abstract: We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to various graph constants. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube.

Patchworking singular algebraic curves II

Abstract: In this paper we present a general patchworking procedure for the construction of reduced singular curves having prescribed singularities and belonging to a given linear system on algebraic surfaces. It originates in the Viro “gluing” method for the construction of real non-singular algebraic hypersurfaces. The general procedure includes almost all known particular modifications, and goes far beyond. Some applications and examples illustrate the construction.

Lie derivations of certain CSL algebras

Abstract: It is shown that each Lie derivation on a reflexive algebra, whose lattice is completely distributive and commutative, can be uniquely decomposed into the sum of a derivation and a linear mapping with image in the center of the algebra.

Banach spaces and groups - order properties and universal models

Abstract: We deal with two natural examples of almost-elementary classes: the class of all Banach spaces (over ℝ or ℂ) and the class of all groups. We show that both of these classes do not have the strict order property, and find the exact place of each one of them in Shelah’sSOP n (strong order property of ordern) hierarchy. Remembering the connection between this hierarchy and the existence of universal models, we conclude, for example, that there are “few” universal Banach spaces (under isometry) of regular density characters.

Density theorems and extremal hypergraph problems

Abstract: We present alternative proofs of density versions of some combinatorial partition theorems originally obtained by E. Szemeredi, H. Furstenberg and Y. Katznelson. These proofs are based on an extremal hypergraph result which was recently obtained independently by W. T. Gowers and B. Nagle, V. Rodl, M. Schacht, J. Skokan by extending Szemeredi’s regularity lemma to hypergraphs.

The horospherical geometry of surfaces in Hyperbolic 4-space

Abstract: We study some geometrical properties associated to the contacts of surfaces with hyperhorospheres inH + 4 (−1). We introduce the concepts of osculating hyperhorospheres, horobinormals, horoasymptotic directions and horospherical points and provide conditions ensuring their existence. We show that totally semiumbilical surfaces have orthogonal horoasymptotic directions.

Polynomial and regular images of R n

Abstract: We obtain new necessary conditions for an n-dimensional semialgebraic subset of R-n to be a polynomial image of R-n. Moreover, we prove that a large family of planar bidimensional semialgebraic sets with piecewise linear boundary are images of polynomial or regular maps, and we estimate in both cases the dimension of their generic fibers.

Schrödinger’s cat

Abstract: We show that the classical framework of probability spaces, which does not admit a model-theoretical treatment, is equivalent to that of probability algebras, which does. We prove that the category of probability algebras is a stable cat, where non-dividing coincides with the ordinary notion of independence used in probability theory.

Ball-covering property of banach spaces

Abstract: We consider the following question: For a Banach spaceX, how many closed balls not containing the origin can cover the sphere of the unit ball? This paper shows that: (1) IfX is smooth and with dimX=n<∞, in particular,X=R n,then the sphere can be covered byn+1 balls andn+1 is the smallest number of balls forming such a covering. (2) Let Λ be the set of all numbersr>0 satisfying: the unit sphere of every Banach spaceX admitting a ball-covering consisting of countably many balls not containing the origin with radii at mostr impliesX is separable. Then the exact upper bound of Λ is 1 and it cannot be attained. (3) IfX is a Gateaux differentiability space or a locally uniformly convex space, then the unit sphere admits such a countable ball-covering if and only ifX * isw *-separable.