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Showing papers in "Israel Journal of Mathematics in 2000"


Journal ArticleDOI
TL;DR: In this paper, the authors considered the limits of the uniform spanning tree and the loop-erased random walk (LERW) on a fine grid in the plane, as the mesh goes to zero.
Abstract: The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.

1,209 citations


Journal ArticleDOI
TL;DR: For subshifts of finite type, conformal repellers, and conformal horseshoes, this article showed that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension.
Abstract: For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.

372 citations


Journal ArticleDOI
TL;DR: In this paper, a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical system of infinite topological dimensions and entropy.
Abstract: In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.

361 citations


Book ChapterDOI
TL;DR: In this article, the authors construct a measure supported on partially hyperbolic sets of diffeomorphisms, where the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting, and the complementary subbundle is non-uniformly expanding.
Abstract: We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms — the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting — under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle).

213 citations


Journal ArticleDOI
TL;DR: In this paper, the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities were investigated and the key point is the establishment of a spectral gap in the spectrum of the transfer operator.
Abstract: We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach.

192 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that as n → ∞ at least 50% of the values L(½, f) are positive, with f varying among the holomorphic new forms of a fixed even integral weight for Γ 0(N) and whose functional equations are even, and that any improvement of 50% is intimately connected to Landau-Siegel zero.
Abstract: We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varying among the holomorphic new forms of a fixed even integral weight for Γ0(N) and whose functional equations are even, are positive. Furthermore, we show that any improvement of 50% is intimately connected to Landau-Siegel zeros. These results may also be used to show that X0(N) = Γ0(N)\ℍ has large quotients with only finitely many rational points. The results below were announced at the conference “Exponential sums” held in Jerusalem, January 1998. The complete proofs, which were presented in courses at Princeton (1997), are being prepared for publication.

187 citations


Journal ArticleDOI
TL;DR: In this paper, a complete isometric representation of the quotient algebra F>>\s ∞/J on Hilbert spaces where J is anywcffff *-closed, 2-sided ideal of F¯¯ ∞, is obtained and used to construct a continuous, F¯¯ *-continuous functional calculus associated to row contractions.
Abstract: General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF ∞ (resp., noncommutative disc algebraA n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF ∞/J on Hilbert spaces whereJ is anyw *-closed, 2-sided ideal ofF ∞, are obtained and used to construct aw *-continuous,F ∞/J-functional calculus associated to row contractionsT=[T 1,…,T n] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF ∞/J are considered.

135 citations


Journal ArticleDOI
TL;DR: In this article, the second step in the proof of existence of equilibrium payoffs for two-player stochastic games was taken, with the case of positive absorbing recursive games.
Abstract: This paper contains the second step in the proof of existence of equilibrium payoffs for two-player stochastic games. It deals with the case of positive absorbing recursive games

118 citations


Journal ArticleDOI
TL;DR: In this paper, the authors study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropics characterization for appropriate measures.
Abstract: LetK be a convex body in ℝn and letWi(K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies which play an important role in the local theory may be described in terms of classical convexity as isotropic ones. We provide new applications of this point of view for the minimal mean width position.

111 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the Poincare series of a geometrically finite group Γ of isometries of a riemannian manifold X with pinched curvature, in the case when Γ contains parabolic elements.
Abstract: In this paper, we study the behaviour of the Poincare series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

110 citations


Journal ArticleDOI
Amnon Besser1
TL;DR: In this article, a new syntomic cohomology for smooth schemes over the ring of integers of ap-adic field is presented, which is more refined than previous constructions and naturally maps to most of them.
Abstract: We construct a new version of syntomic cohomology, called rigid syntomic cohomology, for smooth schemes over the ring of integers of ap-adic field. This version is more refined than previous constructions and naturally maps to most of them. We construct regulators fromK-theory into rigid syntomic cohomology. We also define a “modified” syntomic cohomology, which is better behaved in explicit computations yet is isomorphic to rigid syntomic cohomology in most cases of interest.

Journal ArticleDOI
TL;DR: In this paper, the DH Indistinguishability (DHI) assumption was investigated in the context of double exponential sums, and the authors obtained an upper bound on the statistical distance from uniform is exponentially small.
Abstract: Let p be a large prime such that p − 1 has some large prime factors, and let ϑ ∈ ℤ * be an r-th power residue for all small factors of p−1 The corresponding Diffie-Hellman (DH) distribution is (ϑ x , ϑ y , ϑ xy ) where x, y are randomly chosen from ℤ * A recently formulated assumption is that given p, ϑ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen randomly from ℤ * This assumption, called the DH Indistinguishability (DHI) assumption, turns out to be quite useful and central in cryptography In an effort to investigate the validity of this assumption, we study some statistical properties of DH distributions Let ϑ be an element in ℤ * with sufficiently high multiplicative order We show that if one takes a positive (but sufficiently small) proportion of the most significant bits of each of ϑ x , ϑ y , ϑ xy then one obtains a distribution whose statistical distance from uniform is exponentially small A similar result holds with respect to the least significant bits of (ϑ x , ϑ y , ϑ xy ) We also show somewhat weaker bounds with respect to arbitrary subsets of bit-positions This remarkable property may help gaining assurance in the DHI assumption Our techniques are mainly number-theoretic We obtain an upper bound for double exponential sums with the function aϑ x + bϑ y + cϑ xy which sharpens and generalizes the previous estimates In particular, our bound implies the following result (for p, ϑ of the above form) Ranging over all x, y ∈ ℤ * , the vectors (ϑ x /p, ϑ y /p, ϑ xy /p) are very evenly distributed in the unit cube In order to make this work accessible to two groups of researchers, cryptographers and number theorists, we have decided to make it as self-contained as possible As a result, some parts of it, mainly targetted to one of these groups, may appear obvious to the other In particular we present some basic notions of the modern cryptography and on the other hand we give a short explanation how exponential sums show up in various questions related to uniform distribution of sequences


Journal ArticleDOI
TL;DR: In this article, it was shown that λ to the revised power of κ is the minimal cardinality of a family of subsets of λ each of cardinality κ such that any other subset of the family is included in the union of strictly less than κ members.
Abstract: We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering the known independence results, is “for most pairs κ<λ of regular cardinals, λ to the revised power of κ is equal to λ”. What is the revised power? λ to the revised power of κ is the minimal cardinality of a family of subsets of λ each of cardinality κ such that any other subset of λ of cardinality κ is included in the union of strictly less than κ members of the family. We still have to say what “for most” means. The interpretation we choose is: for every λ, for every large enoughK < ℶw. Under this reinterpretation, we prove the Generalized Continuum Hypothesis.

Journal ArticleDOI
TL;DR: In this paper, the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets was taken, which reduces the existence problem to the class of so-called positive absorbing recursive games.
Abstract: This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper.

Journal ArticleDOI
TL;DR: In this article, Kontsevich [Ko2] and Deligne [De] conjectured a formula for the relation between the two natural products on the space of uni-trivalent diagrams using the related notions of "wheels" and "wheeing".
Abstract: We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT].

Journal ArticleDOI
TL;DR: In this article, the authors considered an Abel equation with polynomials in the plane and gave a center condition that y0 =y(0)≡y(1) for any solutiony(x) of the Abel equation.
Abstract: We consider an Abel equation (*)y’=p(x)y2 +q(x)y3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty0=y(0)≡y(1) for any solutiony(x) of (*).

Journal ArticleDOI
TL;DR: In this paper, it was shown that a Banach space X has the approximation property if and only if the rank operators of norm ≤ 1 are dense in the unit ball of W(Y,X), the space of weakly compact operators from Y to X, in the strong operator topology.
Abstract: Using an isometric version of the Davis, Figiel, Johnson, and Peŀczynski factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.

Journal ArticleDOI
TL;DR: In this article, it was shown that for every odd prime, every k ≥ p ≥ p, and every two subsets, there is a permutationπ ∈S petertodd k>>\s such that the sum of π (i) + π(i) (inZ¯¯¯¯ p>>\s) are pairwise distinct.
Abstract: We prove that for every odd primep, everyk≤p and every two subsets A={a 1, …,a k } andB={b 1, …,b k } of cardinalityk each ofZ p , there is a permutationπ ∈S k such that the sumsa i +b π(i) (inZ p ) are pairwise distinct. This partially settles a question of Snevily. The proof is algebraic, and implies several related results as well.

Journal ArticleDOI
Amnon Besser1
TL;DR: In this article, it was shown that the ad hoc regulator of Coleman and de Shalit for elements of K2 of curves evaluated on a holomorphic differential is the same as the syntomic regulator of the same elements cup produced with the same differential.
Abstract: We show that the ad hoc regulator of Coleman and de Shalit for elements ofK2 of curves evaluated on a holomorphic differential is the same as the syntomic regulator of the same elements cup produced with the same differential. Combined with the results of Coleman and de Shalit this gives a relation between syntomic regulators and special values ofp-adicL-functions. The main technical innovation is the notion of a local index — a kind of generalized residue.

Journal ArticleDOI
Lowell Abrams1
TL;DR: In this paper, it was shown that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then applied this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum co-occurrence of hypersurfaces.
Abstract: The Poincare duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.

Journal ArticleDOI
TL;DR: In this article, a multidimensional generalization of the Agmon-Kannai method is presented for the computation of heat invariants of Riemannian manifolds without boundary.
Abstract: We calculate heat invariants of arbitrary Riemannian manifolds without boundary. Every heat invariant is expressed in terms of powers of the Laplacian and the distance function. Our approach is based on a multidimensional generalization of the Agmon-Kannai method. An application to computation of the Korteweg-de Vries hierarchy is also presented.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the groupoid of Hopf bi-Galois objects, whose objects arek-Hopf algebras, and whose morphisms are L-H-bi-galois extensions ofk for Hopf algesbrasL andH.
Abstract: Letk be a field. We study the groupoid of Hopf bi-Galois objects, whose objects arek-Hopf algebras, and whose morphisms areL-H-bi-Galois extensions ofk for Hopf algebrasL andH. We show that ifH=H N,m is one of the Taft algebras, thenL≅HN,m in anyL-H-bi-Galois object. We compute the group of bi-Galois objects over the two-generator Taft algebrasH N,1. We classify the isomorphism classes of Galois extensions ofk over the general Taft algebrasH N,m, and we compute the groups of bi-Galois objects overH N,m for oddN. Our computations for the two-generator Taft algebras rely on Masuoka's classification [9] of their cleft extensions. To treat the general Taft algebras, we will generalize a result of Kreimer [6] to give a description of the Galois objects over a tensor product of two Hopf algebras.

Journal ArticleDOI
TL;DR: In this paper, it was shown that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑08an4−nan4−n with digits with an ∈ {0, 1,u} has this property.
Abstract: We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑08an4−nan4−n with digits withan ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections.

Journal ArticleDOI
D. Tambara1
TL;DR: In this paper, the tensor categories with fusion rules of self-duality for abelian groups are modeled on the representations of extraspecial 2-groups and the embeddings of those categories are classified into the category of vector spaces, by which the categories are realized as representations of Hopf algebras.
Abstract: The tensor categories with fusion rules of self-duality for abelian groups are modeled on the representations of extraspecial 2-groups. We classify the embeddings of those categories into the category of vector spaces, by which the categories are realized as the representations of Hopf algebras.

Journal ArticleDOI
TL;DR: In this article, the effect of a coagumented idempotent functor on the structure of the objects to which they are applied is investigated. But the effect is limited to groups and not only to fixed groups, but also to non-abelian groups.
Abstract: We consider the effect of a coagumented idempotent functorJ in the the category of groups orG-modules whereG is a fixed group. We are interested in the ‘extent’ to which such functors change the structure of the objects to which they are applied. Some positive results are obtained and examples are given concerning the cardinality and structure ofJ(A) in terms of the cardinality and structure ofA, where the latter is a torsion abelian group. For non-abelian groups some partial results and examples are given connecting the nilpotency classes and the varieties of a groupG andJ(G). Similar but stronger results are obtained in the category ofG-modules.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the normalized cubic Gauss sums for integers c ≡ 1 ((mod 3)) of the field of integers satisfy a uniform distribution around the unit circle for any l ∈ ℤ and any e > 0.
Abstract: It is shown that the normalized cubic Gauss sums for integers c ≡ 1 ((mod 3)) of the field \({\Bbb Q}(\sqrt { - 3} )\) satisfy $${\sum\limits_{N(c) \leqslant X} {\tilde g(c)\Lambda (c)\left( {\frac{c}{{\left| c \right|}}} \right)} ^l} \ll {}_\varepsilon {X^{5/6 + \varepsilon }} + \left| l \right|{X^{3/4 + \varepsilon }},$$ for every l ∈ ℤ and any e > 0 This improves on the estimate established by Heath-Brown and Patterson [4] in demonstrating the uniform distribution of the cubic Gauss sums around the unit circle When l = 0 it is conjectured that the above sum is asymptotically of order X5/6, so that the upper bound is essentially best possible The proof uses a cubic analogue of the author’s mean value estimate for quadratic character sums [3]

Journal ArticleDOI
TL;DR: In this article, the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity was studied, and it was shown that the spacing distribution for square free Q is Poissonian.
Abstract: We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the Bishop-Phelps theorem cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.
Abstract: The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.

Journal ArticleDOI
TL;DR: In this article, the authors used the Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations to study the Rankin-SelbergL-functions of GL k ≥ 2 × classical groups.
Abstract: We use Langlands-Shahidi method and the observation that the local components of residual automorphic representations are unitary representations, to study the Rankin-SelbergL-functions of GL k × classical groups. Especially we prove thatL(s, σ ×τ) is holomorphic, except possibly ats=0, 1/2, 1, whereσ is a cuspidal representation of GL k which satisfies weak Ramanujan property in the sense of Cogdell and Piatetski-Shapiro andτ is any generic cuspidal representation of SO2l+1. Also we study the twisted symmetric cubeL-functions, twisted by cuspidal representations of GL2.