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Showing papers in "Israel Journal of Mathematics in 2003"


Journal ArticleDOI
TL;DR: In this article, a new multifractal formalism for self-similar measures on ℝ with overlaps was proposed, where the authors showed that the variational formula about upper Lyapunov exponents for products of matrices does not hold in this setting.
Abstract: We continue the study in [15, 18] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. In any case, we focus our interest on a special case where the matrix function M(x) takes finite values M 1, ..., M m . In this case, we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [18]. As an application, we prove a new multifractal formalism for a certain class of self-similar measures on ℝ with overlaps. More precisely, let μ be the self-similar measure on ℝ generated by a family of contractive similitudes {S j = ρx + b j } =1 l which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {I j } j∈Λ with disjoint interiors, such that μ is supported on ⋃ j∈Λ I j and the restricted measure $$ \mu |_{I_j } $$ of μ on each interval I j satisfies the complete multifractal formalism. Moreover, the dimension spectrum dim H $$ E_{\mu |_{I_j } } $$ (α) is independent of j.

117 citations


Journal ArticleDOI
TL;DR: In this article, the authors generalize Merzlyakov's theorem on the existence of a formal solution associated with a positive sentence to a general AE sentence which is known to be true over some variety, and then develop tools that enable them to analyze the collection of all such formal solutions.
Abstract: This paper is the second in a series on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the second paper we generalize Merzlyakov’s theorem on the existence of a formal solution associated with a positive sentence [Me]. We first construct a formal solution to a generalAE sentence which is known to be true over some variety, and then develop tools that enable us to analyze the collection of all such formal solutions.

90 citations


Journal ArticleDOI
TL;DR: In this paper, the Fourier transform arguments are replaced by geometric almost orthogonality arguments, and the results in [9] and [10] are extended to multi-dimensional averages.
Abstract: In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments.

77 citations


Journal ArticleDOI
David Galvin1
TL;DR: For the set of homomorphisms from {0, 1} d to Z which send 0 to 0, this paper gave asymptotic formulae for |F| and |F |.
Abstract: WriteF for the set of homomorphisms from {0, 1} d toZ which send0 to 0 (think of members ofF as labellings of {0, 1} d in which adjacent strings get labels differing by exactly 1), andF 1 for those which take on exactlyi values. We give asymptotic formulae for |F| and |F|.

63 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied Lagrangian systems on a closed manifold and showed that the differentiability of Mather's β-function is forced by the irrationality of the homology class.
Abstract: We study Lagrangian systems on a closed manifoldM. We link the differentiability of Mather’sβ-function with the topological complexity of the complement of the Aubry set. As a consequence, whenM is a closed, orientable surface, the differentiability of theβ-function at a given homology class is forced by the irrationality of the homology class. This allows us to prove the two-dimensional case of a conjecture by Mane.

57 citations


Journal ArticleDOI
TL;DR: In this article, the authors provide some nontrivial sets of subgroups of a finite group which are simultaneously G-covering subgroup systems for the classes of supersoluble and nilpotent groups.
Abstract: LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups.

56 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that if a compact manifold has a faithful boundary action of some locally compact group, then the group, the space, and the action can be split into a direct product of a semi-simple Lie group acting on the manifold and a boundary action by a discrete countable group.
Abstract: Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie groupG on homogeneous spacesG/Q, whereQ ⊂G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that ifX is a compact manifold with a faithful boundary action of some locally compact groupH, then (under some mild regularity assumption) the groupH, the spaceX, and the action split into a direct product of a semi-simple Lie groupG acting onG/Q and a boundary action of a discrete countable group.

52 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that any self-similar arc of dimension greater than 1 is a Whitney set, and that any subarc of it fails to be at-quasi-arc for anyt ≥ 1.
Abstract: We study in this paper some relations among self-similar arcs, Whitney sets and quasi-arcs: we prove that any self-similar arc of dimension greater than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to be a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be at-quasi-arc for anyt ≥ 1, which answers an open question on Whitney sets. We also show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent.

46 citations


Journal ArticleDOI
TL;DR: In this article, a one-stage three-player game of incomplete information played on a sequence space {0, 1} where each player has only two moves, the payoff matrix is determined by the 0-coordinate, and all three players know that part of their payoff matrix pertaining to their own payoffs are known.
Abstract: We present an example of a one-stage three-player game of incomplete information played on a sequence space {0, 1} Z such that the players’ locally finite beliefs are conditional probabilities of the canonical Bernoulli distribution on {0, 1} Z , each player has only two moves, the payoff matrix is determined by the 0-coordinate and all three players know that part of the payoff matrix pertaining to their own payoffs. For this example there are many equilibria (assuming the axiom of choice) but none that involve measurable selections of behavior by the players. By measurable we mean with respect to the completion of the canonical probability measure, e.g., all subsets of outer measure zero are measurable. This example demonstrates that the existence of equilibria is also a philosophical issue.

45 citations


Journal ArticleDOI
TL;DR: In this article, the authors characterize the class of functions which occur as the entropy function defined on the set of invariant measures of a (minimal) topological dynamical system.
Abstract: We characterize the class of functions which occur as the entropy function defined on the set of invariant measures of a (minimal) topological dynamical system. Namely, these are all non-negative affine functionsh, defined on metrizable Choquet simplices, which are non-decreasing limits of upper semi-continuous functions. Ifh is itself upper semi-continuous then it can be realized as the entropy function in an expansive dynamical system. The constructions are done effectively using minimal almost 1-1 extensions over a rotation of a group ofp-adic integers (in the expansive case, the construction leads to Toeplitz flows).

43 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of preserving various versions of completeness in (<κ)-support iterations of forcing notions was studied, generalizing the case that S-complete proper is preserved by CS iterations for a stationary costationary.
Abstract: We deal with the problem of preserving various versions of completeness in (<κ)-support iterations of forcing notions, generalizing the case “S-complete proper is preserved by CS iterations for a stationary costationaryS⊆ω1”. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinalκ and a stationary setS⊆κ with fat complement we can have uniformization for every (Aδ:δ ∈S′),Aδ ⊆δ = supAδ, cf (δ) = otp(Aδ) and a stationary non-reflecting setS′⊆S (see B.8.2).

Journal ArticleDOI
TL;DR: In this paper, the authors studied the dynamics of a class of interval translation maps on three intervals and showed that the typical ITM is of finite type (reduce to an interval exchange transformation) and the complement contains a Cantor set.
Abstract: We study the dynamics of a class of interval translation map on three intervals. We show that in this class the typical ITM is of finite type (reduce to an interval exchange transformation) and that the complement contains a Cantor set. We relate our maps to substitution subshifts. Results on Hausdorff dimension of the attractor and on unique ergodicity are obtained.

Journal ArticleDOI
TL;DR: This article showed that the reflection equation (RE) algebra is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra.
Abstract: We prove that the reflection equation (RE) algebraL R associated with a finite dimensional representation of a quasitriangular Hopf algebraH is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show thatL R is a module algebra over the twisted tensor square $${\mathcal{H}}\mathop \otimes \limits^{\mathcal{R}} {\mathcal{H}}$$ and the double D( $${\mathcal{H}}$$ ). We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.

Journal ArticleDOI
TL;DR: In this article, it was shown that a measuredG-space (X, μ), whereG is a locally compact group, is amenable in the sense of Zimmer if and only if the associated unitary representationπ X ofG into L 2(X,μ) is weakly contained into the regular representationλ G and there exists a G-equivariant norm one projection fromL∞(X×X) onto L∞ (X).
Abstract: We show that a measuredG-space (X, μ), whereG is a locally compact group, is amenable in the sense of Zimmer if and only if the following two conditions are satisfied: the associated unitary representationπ X ofG intoL 2(X, μ) is weakly contained into the regular representationλ G and there exists aG-equivariant norm one projection fromL∞(X×X) ontoL∞(X). We give examples of ergodic discrete group actions which are not amenable, althoughπ X is weakly contained intoλ G.

Journal ArticleDOI
TL;DR: In this article, the authors prove inequalities about the quermassintegrals of a convex body in the Euclidean unit ball for a given pair of convex bodies.
Abstract: We prove inequalities about the quermassintegralsV k (K) of a convex bodyK in ℝ n (here,V k (K) is the mixed volumeV((K, k), (B n ,n − k)) whereB n is the Euclidean unit ball) (i) The inequality $$\frac{{V_k \left( {K + L} \right)}}{{V_{k - 1} \left( {K + L} \right)}} \geqslant \frac{{V_k \left( K \right)}}{{V_{k - 1} \left( K \right)}} + \frac{{V_k \left( L \right)}}{{V_{k - 1} \left( L \right)}}$$ holds for every pair of convex bodiesK andL in ℝ n if and only ifk=2 ork=1 (ii) Let 0≤k≤p≤n Then, for everyp-dimensional subspaceE of ℝ n , $$\frac{{V_{n - k} \left( K \right)}}{{\left| K \right|}} \geqslant \frac{1}{{\left( {_{n - p}^{n - p + k} } \right)}}\frac{{V_{p - k} \left( {P_E K} \right)}}{{\left| {P_E K} \right|}},$$ whereP E K denotes the orthogonal projection ofK ontoE The proof is based on a sharp upper estimate for the volume ratio |K|/|L| in terms ofV n−k (K)/V n−k (L), wheneverL andK are two convex bodies in ℝ n such thatK ⊆L

Journal ArticleDOI
TL;DR: For field F of characteristic notp containing a primitive pth root of unity, the Galois module structure of the group of pth-power classes of K for all cyclic extensions K/F of degreep was determined in this paper.
Abstract: For fieldF of characteristic notp containing a primitivepth root of unity, we determine the Galois module structure of the group ofpth-power classes ofK for all cyclic extensionsK/F of degreep.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the expansion of derivatives along orbits of real and complex one-dimensional maps, whose Julia set f attracts a finite set of non-flat critical points.
Abstract: We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ f attracts a finite setCrit of non-flat critical points. Assuming that for eachceCrit, either |D f n(f(c))|→∞ (iff is real) orb n·|Df n(f(c))|→∞ for some summable sequence {b n} (iff is complex; this is equivalent to summability of |D f n(f(c))|−1), we show that for everyxeJ f\U i f −i(Crit), there existl(x)≤max c l(c) andK′(x)>0 $$|Df^n (x)^{l(x)} \ge K^1 (x).\prod\limits_{i = 0}^{s - 1} {(K_i .|} Df^{n_i - n_i + 1} (f(c_i ))|)$$ for infinitely manyn. Here 0=n s<…0 and λ>1. If allceCrit have the same critical order, thenK′(x) is uniformly bounded away from 0. Several corollaries are derived. In the complex case, eitherJ f= $$\hat C$$ orJ f has zero Lebesgue measure. Also (assuming all critical points have the same order) there existk>0 such that ifn is the smallest integer such thatx enters a certain critical neighbourhood, then |Df n(x)|≥k.

Journal ArticleDOI
TL;DR: In this article, the concept of Gorenstein morphism was introduced, and it was shown that a GDLG can be a differential-graded algebra if and only if it is a GGLG.
Abstract: We propose a definition of Gorenstein Differential Graded Algebra. In order to give examples, we introduce the technical notion of Gorenstein morphism. This enables us to prove the following: Theorem:Let A be a noetherian local commutative ring, let L be a bounded complex of finitely generated projective A-modules which is not homotopy equivalent to zero, and let ɛ=Hom A (L, L)be the endomorphism Differential Graded Algebra of L. Then ɛ is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. Theorem:Let A be a noetherian local commutative ring with a sequence of elements a=(a 1,…,a n )in the maximal ideal, and let K(a)be the Koszul complex of a.Then K(a)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. Theorem:Let A be a noetherian local commutative ring containing a field k, and let X be a simply connected topological space with dim k H*(X;k)<∞,which has poincare duality over k. Let C*(X;A)be the singular cochain Differential Graded Algebra of X with coefficients in A. Then C*(X; A)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. The second of these theorems is a generalization of a result by Avramov and Golod from [4].

Journal ArticleDOI
TL;DR: In this paper, the authors introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex Hull of this set.
Abstract: We introduce a notion which is intermediate between that of taking thew*-closed convex hull of a set and taking the norm closed convex hull of this set. This notion helps to streamline the proof (given in [FLP]) of the famous result of James in the separable case. More importantly, it leads to stronger results in the same direction. For example: 1. AssumeX is separable and non-reflexive and its unit sphere is covered by a sequence of balls\(\left\{ {C_i } \right\}_{i = 1}^\infty \) of radiusa<1. Then for every sequence of positive numbers\(\left\{ {\varepsilon _i } \right\}_{i = 1}^\infty \) tending to 0 there is anf eX*, such that ‖f‖ = 1 andf (x)≤1 −ei, wheneverx eCi,i=1,2,… 2. AssumeX is separable and non-reflexive and letT:Y →X* be a bounded linear non-surjective operator. Then there is anf eX* which does not attain its norm onBX such thatf ∉T(Y).

Journal ArticleDOI
TL;DR: In this article, it was shown that certain small sets are removable for bounded mappings of finite distortion for which the distortion function satisfies a suitable subexponential integrability condition.
Abstract: We show that certain small sets are removable for bounded mappings of finite distortion for which the distortion function satisfies a suitable subexponential integrability condition. We also give an example demonstrating the sharpness of this condition.

Journal ArticleDOI
Yuri Medvedev1
TL;DR: Wilson and Zelmanov as discussed by the authors proved that every compact Engel group is locally nil-potent, and they proved the stronger result that every Engel group can be locally nilpotent.
Abstract: In 1992, Wilson and Zelmanov proved that a profinite Engel group is locally nilpotent. Here we prove the stronger result that every compact Engel group is locally nilpotent.

Journal ArticleDOI
TL;DR: In this paper, a large class of groups of isometries of the d-dimensional hyperbolic space is described, which are non-geometrically finite but their Patterson-Sullivan measure is always finite.
Abstract: In this paper, we describe a large class of groups of isometries of thed-dimensional hyperbolic space. These groups may be non-geometrically finite but their Patterson-Sullivan measure is always finite.

Journal ArticleDOI
TL;DR: In this article, the asymptotic equalities of the standard polynomial and the k-th Capelli polynomials over a field F of characteristic zero were studied.
Abstract: Let {c n (St k )} and {c n (C k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold: $$\begin{gathered} c_n \left( {St_{2k} } \right) \simeq c_n \left( {C_{k^2 + 1} } \right) \simeq c_n \left( {M_k \left( F \right)} \right), \hfill \\ c_n \left( {St_{2k + 1} } \right) \simeq c_n \left( {M_{k \times 2k} \left( F \right) \oplus M_{2k \times k} \left( F \right)} \right), \hfill \\ \end{gathered} $$ whereM k (F) is the algebra ofk×k matrices andM k×l (F) is the algebra of (K+l)×(k+l) matrices having the lastl rows and the lastk columns equal to zero. The precise asymptotics ofc n (M k (F)) are known and those ofM k×2k (F) andM 2k×k (F) can be easily deduced. For Capelli polynomials we show that also upper block triangular matrix algebras come into play.

Journal ArticleDOI
TL;DR: In this paper, a class of combinatorial hypersurfaces in the complex projective space is introduced, resulting from non-convex subdivisions of convex polytopes.
Abstract: We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP n and are topologically “glued” out of algebraic hypersurfaces in (ℂ*) n . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.

Journal ArticleDOI
TL;DR: In this paper, generalized Lie bialgebroids over a single point were studied, and it was shown that they can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structure.
Abstract: We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras and we prove that they can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structure. We also propose a method generalizing the Yang-Baxter equation method to obtain generalized Lie bialgebras. Finally, we classify the compact generalized Lie bialgebras.

Journal ArticleDOI
TL;DR: In this article, the authors present some results related to theorems of Pasynkov and Torunczyk on the geometry of maps of finite dimensional compacta, which are related to our results.
Abstract: We present some results related to theorems of Pasynkov and Torunczyk on the geometry of maps of finite dimensional compacta

Journal ArticleDOI
TL;DR: In this article, two-sided estimates of the shape preserving widths were obtained for a finite interval andr ∈ ℕ. Fors=3,…,r+1, r+1.
Abstract: LetI be a finite interval andr ∈ ℕ. Denote by △ + L q the subset of all functionsy ∈L q such that thes-difference △ y(·) is nonnegative onI, ∀τ>0. Further, denote by △ + W the class of functionsx onI with the seminorm ‖x (r) ‖L p ≤1, such that △ x≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the shape preserving widths $$d_n \left( {\Delta _ + ^s W_p^r ,\Delta _ + ^s L_q } \right) = \begin{array}{*{20}c} {\inf } \\ {M^n \in M^n } \\ \end{array} \begin{array}{*{20}c} {\sup } \\ {x \in \Delta _ + ^s W_p^r } \\ \end{array} \begin{array}{*{20}c} {\inf } \\ {y \in M^n \cap \Delta _ + ^s L_q } \\ \end{array} \left\| {x - y} \right\|L_q $$ , whereM n is the set of all linear manifoldsM n inL q , dimM n ≤n, such thatM n ⋂△ + L q ≠ 0.

Journal ArticleDOI
TL;DR: In this article, the case of a successor of the singular cardinal is considered, and the problem of finding the complete cardinal is studied, bearing in mind problems on Whitehead groups, uniformizations and the general problem.
Abstract: On the one hand, we deal with (

Journal ArticleDOI
TL;DR: In particular, the authors showed that well-partial ordering always has a well-ordered linear extension, which is a central result in the theory of partial orderings, allowing one to define, for instance, the dimension of a partial ordering.
Abstract: Szpilrajn’s Theorem states that any partial orderP=〈S,

Journal ArticleDOI
TL;DR: A class of graded simple associative algebras is constructed in this paper, and from them, simple Lie color algesas are obtained, and the structure of these simple Lie colour algesa is explicitly described.
Abstract: A class of graded simple associative algebras are constructed, and from them, simple Lie color algebras are obtained. The structure of these simple Lie color algebras is explicitly described. More precisely, for an (e, Γ)-color-commutative associative algebraA with an identity element over a fieldF of characteristic not 2, and for a color-commutative subalgebraD of color-derivations ofA, denote byA[D] the associative subalgebra of End (A) generated byA (regarded as operators onA via left multiplication) andD. It is easily proved that, as an associative algebra,A[D] is Γ-graded simple if and only ifA is Γ-gradedD-simple. SupposeA is Γ-gradedD-simple. Then, (a)A[D] is a free leftA-module; (b) as a Lie color algebra, the subquotient [A[D],A[D]]/Z(A[D])∩[A[D],A[D]] is simple (except one minor case), whereZ(A[D]) is the color center ofA[D].