Showing papers in "Israel Journal of Mathematics in 1997"

The hardy-littlewood maximal function of a sobolev function

Abstract: We prove that the Hardy-Littlewood maximal operator is bounded in the Sobolev spaceW 1,p (R n ) for 1

Intrinsic ergodicity of smooth interval maps

Abstract: We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy. In this way, we reduce the study ofC r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC ∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.

On embedding expanders into ℓp spaces

Abstract: In this note we show that the minimum distortion required to embed alln-point metric spaces into the Banach space lp is between (c1/p) logn and (c2/p) logn, wherec2>c1>0 are absolute constants and 1≤p

Nearly perfect matchings in regular simple hypergraphs

Abstract: For every fixedk≥3 there exists a constantc k with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc k ND −1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc 3 ND −1/2ln3/2 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N 1/2ln3/2 N), improving results by various authors.

Projecting the one-dimensional Sierpinski gasket

Abstract: LetS⊂ℝ2 be the Cantor set consisting of points (x,y) which have an expansion in negative powers of 3 using digits {(0,0), (1,0), (0,1)}. We show that the projection ofS in any irrational direction has Lebesgue measure 0. The projection in a rational directionp/q has Hausdorff dimension less than 1 unlessp+q ≡ 0 mod 3, in which case the projection has nonempty interior and measure 1/q. We compute bounds on the dimension of the projection for certain sequences of rational directions, and exhibit a residual set of directions for which the projection has dimension 1.

Dynamical properties of expansive one-sided cellular automata

Abstract: To every one-sided one-dimensional cellular automatonF with neighbourhood radiusr we associate its canonical factor defined by considering only the firstr coordinates of all the images of points under the powers ofF. Whenever the cellular automaton is surjective, this factor is a subshift which plays a primary role in its dynamics. In this article we study the class of positively expansive one-sided cellular automata, i.e. those that are conjugate to their canonical factors. This class is a natural generalisation of the toggle or permutative cellular automata introduced in [He]. We prove that the canonical factors of positively expansive one-sided cellular automata are mixing subshifts of finite type that are shift equivalent to full shifts. Moreover, the uniform Bernoulli measure is the unique measure of maximal entropy forF. Consequently, their natural extensions are Bernoulli. We also describe a family of non-permutative positively expansive cellular automata.

On the automorphic theta representation for simply laced groups

Abstract: We construct an automorphic realization of the minimal representation of a split, simply laced groupG, over a number field. The realization is by a residue, at a certain point, of an Eisenstein series, induced from the Borel subgroup. This residue representation is square integrable and defines the automorphic theta representation. It has “very few” Fourier coefficients, which turn out to have some extra invariance properties.

Measure-theoretic complexity of ergodic systems

Abstract: We define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small $$\bar d$$ -balls aroundα-n-names necessary to cover most of the system, for any generating partitionα. We show that this rate is essentially bounded if and only if the system is a translation of a compact group, and compute it for several classes of systems of entropy zero, thus getting examples of growth rates inO(n),O(n k ) fork e ℕ, oro(f(n)) for any given unboundedf, and of various relationships with the usual notion of language complexity of the underlying topological system.

K -groups associated with substitution minimal systems

Abstract: Two ordered Bratteli diagrams can be constructed from an aperiodic substitution minimal dynamical system. One, the proper diagram, has a single maximal path and a single minimal path and the Vershik map on the path space can be extended homeomorphically to a map conjugate to the substitution system. The other, the improper diagram, encodes the substitution more naturally but often has many maximal and minimal paths and no continuous compact dynamics. This paper connects the two diagrams by considering theirK 0-groups, obtaining the equation $$K_0 (Proper) = K_0 (Improper)/Q \oplus \mathbb{Z}^ u $$ whereQ and ν can be determined from the combinatorial properties of the substitution. This allows several examples of substitution sequences to be distinguished at the level of strong orbit equivalence. A final section shows that every dimension group with unit which is a stationary limit of ℤ n groups can be represented as aK 0 group of some substitution minimal system. Also every stationary proper minimal ordered Bratteli diagram has a Vershik map which is either Kakutani equivalent to ad-adic system or is conjugate to a substitution minimal system. The equation above applies to a much wider class which includes those minimal transformations which can be represented as a path-sequence dynamical system on a Bratteli diagram with a uniformly bounded number of vertices in each level.

Interaction of legendre curves and lagrangian submanifolds

Abstract: It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form Mn(4c), n >_ 2, except the totally geodesic ones. In this paper we introduce the notion of Lagrangian Humbilical submanifolds which are the "simplest" Lagrangian submanifolds next to the totally geodesic ones in complex-space-forms. We show that for each Legendre curve in a 3-sphere S 3 (respectively, in a 3-dimensional antide Sitter space-time H3), there associates a Lagrangian H-umbilical submanifold in CP n (respectively, in CH n ) via warped products. The main part of this paper is devoted to the classification of Lagrangian H-umbi|ical submanifolds in CP n and in C/'/n . Our classification theorems imply in particular that "except some exceptional classes", Lagrangian H-umbilical submanlfolds of CP n and of CH n axe obtained from Legendre curves in S 3 or in/-13 via warped products. This provides us an interesting interruption of Legendre curves and Lagrangian H-umbilicM submanifolds in non-fiat complex-space-forms. As an immediate by-product, our results provide us many concrete examples of Lagrangian H-umbilical isometric immersions of real-space-forms into non-fiat complex-space-forms.

Crossed modules and doi-hopf modules

Abstract: We prove that crossed modules (or Yetter-Drinfel’d modules) are special cases of Doi’s unified Hopf modules. The category of crossedH-modules is therefore a Grothendieck category (if we work over a field), and the Drinfel’d double appears as a type of generalized smash product.

Degenerate principal series representations forU(n, n)

Abstract: For a quadratic extensionE/F of a nonarchimedean local field of characteristic other than 2, letG=U (n, n) be the quasisplit unitary group of rankn, and letP be the maximal parabolic subgroup ofG which stabilizes a maximal isotropic subspace. ThenP has a Levi decompositionP=MN withM ≃ GL (n, E). In this paper, the points of reducibility and composition series of the degenerate principal seriesI n (s, χ) defined by characters ofM are determined completely. The constituents arising as theta lifts of characters ofU (m)'s are identified and their behavior under the intertwining operator $$M(s,\chi ):I_n (s,\chi ) \to I_n (s,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\chi } )$$ ) is described. The caseE=F⊕F andG≃GL (2n, F) is included.

Critical exponent and critical blow-up for quasilinear parabolic equations

Abstract: We consider nonnegative solutions to the Cauchy problem or to the exterior Dirichlet problem for the quasilinear parabolic equationsu t=Δu m+up with 1p m * , then there are nontrivial global solutions (global existence case). In this paper we show: (i) For the Cauchy problem,p * belongs to the blow-up case. (ii) For the exterior Dirichlet problem,p * also gives the critical exponent of blow-up.

Semismall perturbations in the martin theory for elliptic equations

Abstract: We investigate stability of Martin boundaries for positive solutions of elliptic partial differential equations. We define a perturbation which isGLD-semismall at infinity, show that Martin boundaries are stable under this perturbation, and give sufficient conditions for it.

A simple characterization of the set ofμ-entropy pairs and applications

Abstract: We present simple characterizations of the setsE μ andE X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE X . Finally an example of a minimal point distal dynamical system is constructed for whichE X ∩(X 0×X 0)≠ $$ ot 0$$ , whereX 0 is the denseG δ subset of distal points inX.

Neighborly cubical spheres and a cubical lower bound conjecture

Abstract: Using mirrors and cyclic polytopes, we construct cubicald-spheres which are the analogs of cyclic polytopes in the sense that they have the ⌉d−1/2⌈-skeleta of cubes. The existence of these neighborly cubical spheres leads to a special case of an upper bound conjecture for cubical spheres, suggested by Kalai. We extend the same construction to show that the closed convex hull off-vectors of cubical spheres contains a cone described by Adin, as an analog to the generalized lower bound theorem for simplicial polytopes.

A uniformly convex hereditarily indecomposable banach space

Abstract: We construct a uniformly convex hereditarily indecomposable Banach space, using a method similar to the one of Gowers and Maurey in [GM], and the theory of complex interpolation for a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher, and Weiss ([CCRSW1]).

Rank-one weak mixing for nonsingular transformations

Abstract: We construct rank-one infinite measure preserving transformations satisfying each of the following dynamical properties: (1) ContinuousL∞ spectrum, conservativek-fold cartesian products but nonergodic cartesian square; (2) ergodick-fold cartesian products; (3) nonconservative cartesian square. We show how to modify the construction of (1) to obtain type IIIλ transformations with similar properties.

Degenerate principal series and local theta correspondence II

Abstract: Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ω p,q and Ω p,q (1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].

Explicit formulas for the waldspurger and bessel models

Abstract: This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Bocherer and Mizumoto forn=2 and base field ℚ, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.

Einstein, conformally flat and semi-symmetric submanifolds satisfying chen’s equality

Abstract: In a recent paper, B. Y. Chen proved a basic inequality between the intrinsic scalar invariants infK andτ ofM n , and the extrinsic scalar invariant |H|, being the length of the mean curvature vector fieldH ofM n in $$\mathbb{E}^m $$ . In the present paper we classify the submanifoldsM n of $$\mathbb{E}^m $$ for which the basic inequality actually is an equality, under the additional assumption thatM n satisfies some of the most primitive Riemannian curvature conditions, such as to be Einstein, conformally flat or semi-symmetric.

Rank one transformations with singular spectral type

Abstract: We show that a certain class of measures arising from generalized Riesz products is singular. In particular, cutting and stacking (i.e. rank one) transformations whose cuts do not grow too rapidly, have singular maximal spectral type. The precise condition is $$\sum olimits_{n = 1}^\infty {(1/\omega _n^2 )} $$ , wherew h is the number of cuts at stagen.

On the Q-rational cuspidal subgroup and the component group ofJ 0(p r )

Abstract: Forp≥3 a prime, we compute theQ-rational cuspidal subgroupC(p r ) of the JacobianJ 0(p r ) of the modular curveX 0(p r ). This result is then applied to determine the component group Φ p r of the Neron model ofJ 0(p r ) overZ p . This extends results of Lorenzini [7]. We also study the action of the Atkin-Lehner involution on thep-primary part ofC(p r ), as well as the effect of degeneracy maps on the component groups.

Sums of continuous and differentiable functions in dynamical systems

Abstract: LetT be a homeomorphism of a metrizable compactX, the sequencec k/k tends to 0 andc k tends to infinity. We’ll study the limit behaviour of the distributions of the sums (1/c k) ∑ =0 -1 F oT i whereF is from a space of continuous functions—the central limit problem and the speed of convergence in the ergodic theorem. The main attention is given to the case whereX is the unit circle andT is an irrational rotation; in this case we consider the spaces of absolutely continuous, Lipschitz, andk-times differentiable functionsF.

Quantum groups of dimensionpq 2

Abstract: In this paper we construct two families of non-trivial self-dual semisimple Hopf algebras of dimensionpq2 and investigate closely their (quasi) triangular structures. The paper contains also general results on finite-dimensional triangular Hopf algebras, unimodularity, semisimplicity and ribbon structures of finite-dimensional semisimple Hopf algebras.

Guarding galleries where every point sees a large area

Abstract: We prove a conjecture of Kavraki, Latombe, Motwani and Raghavan that ifX is a compact simply connected set in the plane of Lebesgue measure 1, such that any pointx∈X sees a part ofX of measure at least ɛ, then one can choose a setG of at mostconst1/ɛ log 1/ɛ points inX such that any point ofX is seen by some point ofG. More generally, if for anyk points inX there is a point seeing at least 3 of them, then all points ofX can be seen from at mostO(k3 logk) points.

Bounds for fixed point free elements in a transitive group and applications to curves over finite fields

Abstract: We investigateV f , the cardinality of the value set of a polynomialf of degreen over a finite field of cardinalityq It has been shown that iff is not bijective, thenV f ≤q−(q−1)/n Polynomials do exist which essentially achieve that bound We do prove that if the degree off is prime to the characteristic andf is not bijective, then asymptoticallyV f ≤(5/6)q We consider related problems for curves and higher dimensional varieties This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well

The kadec-klee property in symmetric spaces of measurable operators

Abstract: We show that ifE is a separable symmetric Banach function space on the positive half-line thenE has the Kadec-Klee property if and only if, for every semifinite von Neumann algebra (M, τ), the associated spaceE(M, τ) ofτ-measurable operators has the Kadec-Klee property.

Estimates related to sumfree subsets of sets of integers

Abstract: A subsetA of the positive integers ℤ+ is called sumfree provided (A+A)∩A=o. It is shown that any finite subsetB of ℤ+ contains a sumfree subsetA such that |A|≥1/3(|B|+2), which is a slight improvement of earlier results of P. Erdos [Erd] and N. Alon-D. Kleitman [A-K]. The method used is harmonic analysis, refining the original approach of Erdos. In general, defines k (B) as the maximum size of ak-sumfree subsetA ofB, i.e. (A) k = $$\underbrace {A + ... + A}_{k times}$$ % MathType!End!2!1! is disjoint fromA. Elaborating the techniques permits one to prove that, for instance, $$s_3 \left( B \right) > \frac{{\left| B \right|}}{4} + c\frac{{\log \left| B \right|}}{{\log \log \left| B \right|}}$$ % MathType!End!2!1!as an improvement of the estimate $$s_k \left( B \right) > \frac{{\left| B \right|}}{4}$$ % MathType!End!2!1! resulting from Erdos’ argument. It is also shown that in an inequalitys k (B)>δ k |B|, valid for any finite subsetB of ℤ+, necessarilyδ k → 0 fork → ∞ (which seemed to be an unclear issue). The most interesting part of the paper are the methods we believe and the resulting harmonic analysis questions. They may be worthwhile to pursue.

On the gamma factors attached to representations ofU(2,1) over ap-adic field

Abstract: In this paper we complete the local non-archimedean theory of Rankin-Selberg convolutions forU(2,1) that was suggested by S. Gelbart and I. Piatetski-Shapiro in [G,PS,1]. In addition, we prove two fundamental properties of the gamma factors (Theorem 6.3) which allow us to give a new proof of a strong multiplicity one theorem forU(2,1).