Showing papers in "Israel Journal of Mathematics in 2002"
TL;DR: In this paper, the authors give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphere, where the bi-harmonic equation is solved explicitly.
Abstract: We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereS
n
. In the case of curves inS
n
we solve explicitly the biharmonic equation.
249 citations
TL;DR: In this paper, a duality theory between the continuous representations of a compact p-adic Lie group in Banach spaces over a given padic field and certain compact modules over the completed group ring is introduced.
Abstract: We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringoK[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗oK[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL2(ℤp).
157 citations
TL;DR: In this article, it was shown that the probability that the s-th largest eigenvalue of a random symmetric n-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more than four times is at most 4e −¯¯¯¯ t 232 s2.
Abstract: It is shown that for every 1≤s≤n, the probability that thes-th largest eigenvalue of a random symmetricn-by-n matrix with independent random entries of absolute value at most 1 deviates from its median by more thant is at most 4e
−
t
232
s2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces.
150 citations
TL;DR: A general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1}N is obtained.
Abstract: In this paper we obtain a general lower bound for the tail distribution of the Fourier spectrum of Boolean functionsf on {1, −1}N. Roughly speaking, fixingk∈ℤ+ and assuming thatf is not essentially determined by a bounded number (depending onk) of variables, we have that\(\sum {\left| s \right| > k\left| {\hat f(S)} \right|^2 } \gtrsim k^{ - 1/2 - \varepsilon } \). The example of the majority function shows that this result is basically optimal.
145 citations
TL;DR: In this paper, the authors studied compact group extensions of hyperbolic diffeomorphisms and showed that the correlations decay faster than any power of time and that this is always the case for ergodic semisimple extensions.
Abstract: We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds.
125 citations
TL;DR: In this article, the authors further studied the connection between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces, and showed that every topological group is approximated by finite groups in a certain weak sense.
Abstract: In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso
$$\left( \mathbb{U} \right)$$
of isometries of Urysohn’s universal complete separable metric space
$$\mathbb{U}$$
, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if
$$\mathbb{U}$$
is replaced with any generalized Urysohn metric spaceU that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form Iso(U), our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.
111 citations
TL;DR: In this paper, a generalization of Selberg's theorem for infinite index congruence subgroups of SL2(Z) is presented, and for such subgroups with a high enough Hausdorff dimension of the limit set, a spectral gap property is established.
Abstract: A celebrated theorem of Selberg states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. Extending the work of Sarnak and Xue for cocompact arithmetic lattices, we prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). For such subgroups with a high enough Hausdorff dimension of the limit set we establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs.
108 citations
TL;DR: In this article, an elementary argument from abstract ergodic theory was used to prove mixing of hyperbolic flows on negatively curved manifolds, and the mixing property of product measures for geodesic flows on (not necessarily compact) negatively curved manifold was shown.
Abstract: We describe an elementary argument from abstract ergodic theory that can be used to prove mixing of hyperbolic flows. We use this argument to prove the mixing property of product measures for geodesic flows on (not necessarily compact) negatively curved manifolds. We also show the mixing property for the measure of maximal entropy of a compact rank-one manifold.
106 citations
TL;DR: In this article, it was shown that if π is archimedean, and π v is spherical principal series representation with trivial central character, then the local L-factor and local integral are equal.
Abstract: The Rankin-Selberg method associates, to each local factorL(s, π
v
× π
′
) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π
v
and
′
. In this paper we show that, if ν is archimedean, and π
v
and
′
are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one. We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.
96 citations
TL;DR: In this paper, it was shown that polynomial mappings to a nilpotent group form a group with respect to the elementwise multiplication, and that any mapping of a groupG to a groupF splits into a homomorphismG→G' to a non-nilpotent groupsG' and a mappingG'→F'.
Abstract: A mapping ϕ of a groupG to a groupF is said to be polynomial if it trivializes after several consecutive applications of operatorsD
h
,h ∈G, defined byD
h
ϕ(g)=ϕ(g)
−1
ϕ(gh). We study polynomial mappings of groups, mainly to nilpotent groups. In particular, we prove that polynomial mappings to a nilpotent group form a group with respect to the elementwise multiplication, and that any polynomial mappingG→F to a nilpotent groupF splits into a homomorphismG→G’ to a nilpotent groupG’ and a polynomial mappingG’→F. We apply the obtained results to prove the existence of the compact/weak mixing decomposition of a Hilbert space under a unitary polynomial action of a finitely generated nilpotent group.
91 citations
TL;DR: In this paper, the authors obtained new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds and extended previous results of Gromov, Oh, Polterovich and Viterbo to the more general case of subcritical Steins.
Abstract: In this paper we obtain new topological restrictions on Lagrangian embeddings into subcritical Stein manifolds. We also extend previous results of Gromov, Oh, Polterovich and Viterbo on Lagrangian submanifolds of ℂ
n
to the more general case of subcritical Stein manifolds.
TL;DR: In this article, the linearity of braid groups was extended in two ways: 1) by extending the Krammer representation to Artin groups of finite type, and 2) by generalizing the Kramer representation to groups of infinite type.
Abstract: Recent results on the linearity of braid groups are extended in two ways. We generalize the Lawrence Krammer representation as well as Krammer’s faithfulness proof for this linear representation to Artin groups of finite type.
TL;DR: In this article, an elementary formula about the average expansion of certain products of 2 by 2 matrices is given. But it is not shown that this formula holds for all products of spectral radius.
Abstract: We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman’s result. Finally, we give a result about the growth of the spectral radius of products.
TL;DR: In this article, the authors studied partially hyperbolic attractors of C 2 diffeomorphisms on a compact manifold and constructed Sinai-Ruelle-Bowen measures, for which they proved exponential decay of correlations and the central limit theorem, in the space of Holder continuous functions.
Abstract: We study partially hyperbolic attractors ofC
2 diffeomorphisms on a compact manifold. For a robust (non-empty interior) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of Holder continuous functions. The techniques we develop (backward inducing, redundancy elimination algorithm) should be useful in the study of the stochastic properties of much more general non-uniformly hyperbolic systems.
TL;DR: In this article, the problem of uniformization of general Riemann surfaces through consideration of the curvature equation was considered, and in particular of constructing Poincare metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e2u=Ko(z).
Abstract: We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincare metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e2u=Ko(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore2u giving the Poincare metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem.
TL;DR: In this paper, a uniquely ergodic Toeplitz flow was constructed for the case that each cylinder set is a cylinder set, with the condition that the cylinder set be a cylinder.
Abstract: LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ
0
∞
G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ
U
(x)=inf{k⩾1:T
k
xeU}, and defineG
U
(t)=1/u(U)u({xeU:u(U)τU(x)>t),teℝ We prove that for μ-a.e.x∈X, there exists a sequence (U
n
)
n≥1
of neighbourhoods ofx such that {x}=∩
n
U
n
, and for anyG ∈G, there exists a subsequence (n
k
)
k≥1
withG
U
n
k
↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x
∞,S, μ), the orbit closure of a Toeplitz sequencex
∞, such that the above conclusion still holds, with moreover the requirement that eachU
n
be a cylinder set.
TL;DR: In this article, the normal forms of germs at (0, 0) of Poisson structures on G2 (G=ℝ or ℂ) and recall a result given by Arnold.
Abstract: It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures onG2 (G=ℝ or ℂ) and recall a result given by Arnold. Then we will compute locally the Poisson cohomology of a particular type of Poisson structure.
TL;DR: In this article, the authors prove the functoriality of the analytic torsion forms of Bismut and Lott [BLo] with respect to the composition of two submersions.
Abstract: In this paper, we prove the functoriality of the analytic torsion forms of Bismut and Lott [BLo] with respect to the composition of two submersions.
TL;DR: In this paper, it was shown that the nilpotent class of finite groups with only two conjugacy lengths is at most 3, which corresponds to the result of Isaacs and Passman for degrees of irreducible characters.
Abstract: We show that the nilpotent class of any finite group which has only two conjugacy lengths is at most 3. This corresponds to a result of Isaacs and Passman for degrees of irreducible characters.
TL;DR: In this article, for certain group extensions of uniquely ergodic transformations, the authors identify all locally finite and invariant invariant measures, and establish the asymptotic behaviour for these group extensions proving logarithmic and bounded rational ergodicity.
Abstract: For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures. These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic ergodic theorems, and bounded rational ergodicity.
TL;DR: In this article, the authors studied 2-graded polynomial identities satisfied by matrix algebra of order two M 2(K), matrix algebra M 1,1(G), and matrix algebra G ⊗ K G. The gradings on these three algebras are supposed to be the standard ones.
Abstract: In this paper we study 2-graded polynomial identities. We describe bases of these identities satisfied by the matrix algebra of order twoM 2(K), by the algebraM 1,1(G), and by the algebraG ⊗ K G. HereK is an arbitrary infinite field of characteristic not 2,G stands for the Grassmann (or exterior) algebra of an infinite dimensional vector space overK, andM 1,1(G) is the algebra of all 2×2 matrices overG whose entries on the main diagonal are even elements ofG, and those on the second diagonal are odd elements ofG. The gradings on these three algebras are supposed to be the standard ones.
TL;DR: In this paper, the Gauss measure is used as a weak limit of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state for a continuous real valued function if and only if dμ < ∞.
Abstract: We give a general definition of the topological pressureP
top
(f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if ef
−
dμ<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.
TL;DR: In this article, it was shown that [rn]m (m>2) can be expressed in terms of linearly recurrent sequences with orders not exceeding ϕ(m)/2.
Abstract: In this paper we study [rn]m=Σk≡r(modm) (kn) wherem>0,n≥0 andr are integers. We show that [rn]m (m>2) can be expressed in terms of some linearly recurrent sequences with orders not exceeding ϕ(m)/2. In particular, we determine [rn]12 explicitly in terms of first order and second order recurrences. It follows that for any primep>3 we have\(\frac{{2^{p - 1} - 1}}{p} \equiv 2\left( { - 1} \right)^{\left( {p - 1} \right)/2} \sum\limits_{1 \leqslant k1 \leqslant \left( {p + 1} \right)/6} {\frac{{\left( { - 1} \right)^k }}{{2k - 1}}\left( {\bmod p} \right)} \) and\(\sum\limits_{0< k< p/2} {\frac{{3^k }}{k} \equiv } \sum\limits_{0< k< p/6} {\frac{{\left( { - 1^k } \right)}}{k} \equiv } \left( {\bmod p} \right)\).
TL;DR: In this article, the authors studied the problem of quasi-isomorphic liftings of Nichols algebras of Yetter-Drinfeld modules of Cartan typeB 2 subject to the small restriction that the diagnonal elements of the braiding matrix are primitiventh roots of 1 with oddn≠5.
Abstract: We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan typeB
2 subject to the small restriction that the diagnonal elements of the braiding matrix are primitiventh roots of 1 with oddn≠5. As well, we compute the liftings of a Nichols algebra of Cartan typeA
2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered in previous work of Andruskiewitsch and Schneider. We study the problem of when the liftings of a given Nichols algebra are quasi-isomorphic. The Appendix (with I. Rutherford) contains a generalization of the quantum binomial formula. This formula was used in the computation of liftings of typeB
2 but is also of interest independent of these results.
TL;DR: In this paper, the sharp order estimate for the First Painleve Differential Equation ρ =z+6w 6w 2, commonly known as first painleve transcendents, was obtained.
Abstract: We consider the solutions of the First Painleve Differential Equationω″=z+6w
2, commonly known as First Painleve Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z
−1
w(z
2), respectively:w#(z)=O(|z|3/4) andf#(z)=O(|z|3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painleve’s Equations II and IV.
TL;DR: In this article, the authors studied pre-balanced dualizing complexes over noncommutative complete semilocal algebras and proved an analogue of Van den Bergh's theorem.
Abstract: We study pre-balanced dualizing complexes over noncommutative complete semilocal algebras and prove an analogue of Van den Bergh’s theorem [VdB, 6.3]. The relationship between pre-balanced dualizing complexes and Morita dualities is studied. Some immediate applications to classical ring theory are also given.
TL;DR: In this paper, the mixing properties of equilibrium statesμ of non-Markov piecewise invertible mapsT:X→X, especially in the multidimensional case, were studied.
Abstract: We study the mixing properties of equilibrium statesμ of non-Markov piecewise invertible mapsT:X→X, especially in the multidimensional case. Assuming mainly Holder continuity and that the topological pressure of the boundary is smaller than the total topological pressure, we establish exponential decay of correlations, i.e.,\(\left| {\int_x {\varphi \cdot \psi oT^n d\mu - \int_x {\varphi d\mu \cdot \int_x {\psi d\mu } } } } \right| \leqslant C \cdot e^{ - an} \) for all Holder functionsϕ,ψ :X→ℝ, alln≥0 and someC 0. We also obtain a Central Limit Theorem. Weakening the smoothness assumption, we get subexponential rates of decay.
TL;DR: In this article, a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation was constructed, and the local limit theorem was established for the partition function of the equilibrium measure.
Abstract: We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [5] for the above class of processes. The method used goes back to A. Khintchine.
TL;DR: In this article, it was shown that many lattices in Lie groups do not admit a Kazhdan constant which is independent of the generating subset, which is a question of A. Lubotzky.
Abstract: In this paper we answer a question of A. Lubotzky by giving examples of groups having property (T) without uniform Kazhdan constants. We show that many lattices in Lie groups do not admit a Kazhdan constant which is independent of the generating subset.
TL;DR: In this paper, the authors established conditions under which the Haar measure on the sphere concentrates on a neighborhood of the intersection of the sphere with a subspace of a given dimension.
Abstract: In this paper we establish concentration phenomena for subspaces with arbitrary dimension. Namely, we display conditions under which the Haar measure on the sphere concentrates on a neighborhood of the intersection of the sphere with a subspace ofRn of a given dimension. We display applications to a problem of projections of points on the sphere, and to the duality of entropy numbers conjecture.