Showing papers in "Israel Journal of Mathematics in 2011"

Nonuniform hyperbolicity for c 1 -generic diffeomorphisms

Abstract: We study the ergodic theory of non-conservative C 1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C 1-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C 1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ. In addition, confirming a claim made by R. Mane in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin’s Stable Manifold Theorem, even if the diffeomorphism is only C 1.

Where to place a hole to achieve a maximal escape rate

Abstract: A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the faster is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.

An improved construction of progression-free sets

Abstract: The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turan in 1936. They have presented a construction with $$|S| = \Omega ({n^{{{\log }_3}2}})$$ elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is $$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }} \cdot {{\log }^{1/4}}n}}} \right).$$ Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of $${\rm{\Theta }}\left( {\sqrt {\log n} } \right)$$ , and shows that $$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }}}} \cdot {{\log }^{1/4}}n} \right).$$ In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained. We also present an application of our proof technique in Discrete Geometry.

Supertropical matrix algebra

Abstract: The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows:

Measure-theoretical sensitivity and equicontinuity

Abstract: For an invariant measure µ in a topological dynamics, notions of µ-sensitivity, µ-complexity and µ-equicontinuity are introduced and investigated. It turns out that µ-sensitivity defined here is equivalent to pairwise sensitivity defined by Cadre and Jacob. For an ergodic µ, µ-equicontinuity, no µ-complexity pair and non-µ-sensitivity are equivalent, which implies minimality and equicontinuity when restricted to the support. Moreover, the notion of µ-sensitive set is introduced, it is shown that a transitive system with an ergodic measure of full support has zero topological entropy if there is no uncountable µ-sensitive set, and a non-minimal transitive system with dense minimal points has infinite sequence entropy for some sequence. For a minimal system (X, T) it is shown that (x 1, x 2) is regionally proximal if and only if, for any neighborhood U of x 2, {n∈ℤ+:T n x 1∊U} is a Poincare sequence. This implies that if (x i , x i+1) is regionally proximal for i = 1, …, n − 1, then (x 1, …, x n ) is n-regionally proximal. The structure of a minimal system is determined via the cardinalities of sensitive sets for µ.

The noncommutative Choquet boundary II: Hyperrigidity

Abstract: A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space A ⊆ B(H) and every sequence of unital completely positive linear maps ϕ 1, ϕ 2,... from B(H) to itself, $$\mathop {\lim }\limits_{n \to \infty } ||{\phi _n}(g) - g|| = 0,{\forall _g} \in G \Rightarrow \mathop {\lim }\limits_{n \to \infty } {\phi _n}(a) - a|| = 0,{\forall _a} \in A.$$ We show that one can determine whether a given set G of generators is hyperrigid by examining the noncommutative Choquet boundary of the operator space spanned by G ∪ G*. We present a variety of concrete applications and discuss prospects for further development.

Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds

Abstract: We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its “boundary depth,” and establish basic results about how the boundary depths of different Hamiltonians are related. As applications, we prove that certain Hamiltonian symplectomorphisms supported in displaceable subsets have infinitely many nontrivial geometrically distinct periodic points, and we also significantly expand the class of coisotropic submanifolds which are known to have positive displacement energy. For instance, any coisotropic submanifold of contact type (in the sense of Bolle) in any closed symplectic manifold has positive displacement energy, as does any stable coisotropic submanifold of a Stein manifold. We also show that any stable coisotropic submanifold admits a Riemannian metric that makes its characteristic foliation totally geodesic, and that this latter, weaker, condition is enough to imply positive displacement energy under certain topological hypotheses.

A fourier-type transform on translation-invariant valuations on convex sets

Abstract: Let V be a finite-dimensional real vector space. Let V al sm (V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism $$ \mathbb{F}_V :Val^{sm} (V)\tilde \to Val^{sm} (V^* ) \otimes Dens(V) $$ such that $$ \mathbb{F}_V $$ commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in [5]) to the convolution on the target (introduced in [16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

Torus invariant divisors

Abstract: Using the language of Altmann, Hausen and Sus, we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture, X is given by a divisorial fan on a smooth projective curve Y. Cartier divisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one-dimensional faces of it. Furthermore, we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(Dh) will be determined by a subset of a weight polytope associated to h, and global sections of specific line bundles on the underlying curve Y.

Artin-Schreier extensions in NIP and simple fields

Abstract: We show that NIP fields have no Artin-Schreier extension, and that simple fields have only a finite number of them.

Supertropical matrix algebra ii: solving tropical equations

Abstract: We continue the study of matrices over a supertropical algebra, proving the existence of a tangible adjoint of A, which provides the unique right (resp. left) quasi-inverse maximal with respect to the right (resp. left) quasi-identity matrix corresponding to A; this provides a unique maximal (tangible) solution to supertropical vector equations, via a version of Cramer’s rule. We also describe various properties of this tangible adjoint, and use it to compute supertropical eigenvectors, thereby producing an example in which an n × n matrix has n distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.

Substitution tilings and separated nets with similarities to the integer lattice

Abstract: We show that any primitive substitution tiling of ℝ2 creates a separated net which is biLipschitz to ℤ2 Then we show that if H is a primitive Pisot substitution in ℝd, for every separated net Y, that corresponds to some tiling τ ∈ XH, there exists a bijection Φ between Y and the integer lattice such that supy∈Y∥Φ(y) − y∥ < ∞ As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling The proofs rely on results of Laczkovich, and Burago and Kleiner

Lower bounds for weak epsilon-nets and stair-convexity

Abstract: A set N ⊂ ℝ d is called a weak ɛ-net (with respect to convex sets) for a finite X ⊂ ℝ d if N intersects every convex set C with |X ∩ C| ≥ ɛ|X|. For every fixed d ≥ 2 and every r ≥ 1 we construct sets X ⊂ ℝ d for which every weak 1/r -net has at least Ω(r logd−1 r) points; this is the first superlinear lower bound for weak ɛ-nets in a fixed dimension. The construction is a stretched grid, i.e., the Cartesian product of d suitable fast-growing finite sequences, and convexity in this grid can be analyzed using stair-convexity, a new variant of the usual notion of convexity. We also consider weak ɛ-nets for the diagonal of our stretched grid in ℝ d , d ≥ 3, which is an “intrinsically 1-dimensional” point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that the upper bounds by Alon, Kaplan, Nivasch, Sharir and Smorodinsky (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the known upper bound for the so-called second selection lemma in the plane by a logarithmic factor: We obtain a set T of t triangles with vertices in an n-point set in the plane such that no point is contained in more than O(t 2/(n 3 logn 3/t)) triangles of T.

Some approximation properties of Banach spaces and Banach lattices

Abstract: The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is a ℒ ∞-space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is a ℒ 1-space and Z is a ℒ p -space (1 ≤ p ≤ ∞), then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GL-l.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions of bounded variation on ℝ n .

Model theoretic connected components of groups

Abstract: We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends a theorem of Shelah from [21]. We consider also in this context the multiplicative and the additive groups of some rings (including infinite fields).

Relative Riemann-Zariski spaces

Abstract: In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata’s compactification theorem.

Degenerate principal series representations for quaternionic unitary groups

Abstract: We give a complete description of all points of reducibility and the composition series of the degenerate principal series representations for quaternionic unitary groups which are induced from a character of the maximal parabolic subgroup with abelian unipotent radical. The case of even orthogonal groups is also included.

Integral geometry under G2 and Spin(7)

Abstract: A Hadwiger-type theorem for the exceptional Lie groups G2 and Spin(7) is proved. The algebras of G2 or Spin(7) invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.

On the length of commutators in chevalley groups

Abstract: Let G = G(Φ,R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ,R) its elementary subgroup. The main result of the article asserts that the set of commutators C = {[a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = Al, with a better bound, was earlier obtained by Sivatski and Stepanov.

Quasisymmetric functions and Kazhdan-Lusztig polynomials

Abstract: We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.

Weak isomorphisms between Bernoulli shifts

Abstract: In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.

The socle series of a Leavitt path algebra

Abstract: We investigate the ascending Loewy socle series of Leavitt path algebras LK(E) for an arbitrary graph E and field K. We classify those graphs E for which LK(E) = Sλ for some element Sλ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of LK(E) is λ. Moreover, λ ≤ ω1 (the first uncountable ordinal) if E is a row-finite graph.

Closures of quadratic modules

Abstract: We consider the problem of determining the closure \(\bar M\) of a quadratic module M in a commutative ℝ-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable [28][29] and in analyzing algorithms for polynomial optimization involving semidefinite programming [12]. The closure of a semiordering is also considered, and it is shown that the space \(\mathcal{Y}_M\) consisting of all semiorderings lying over M plays an important role in understanding the closure of M. The result of Schmudgen for preorderings in [29] is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of \(\bar M\) which is valid in many cases.

Finite groups have even more conjugacy classes

Abstract: In his paper Finite groups have many conjugacy classes (J. London Math. Soc (2) 46 (1992), 239–249), L. Pyber proved the to-date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber’s paper.

On the Schur multipliers of finite p-groups of given coclass

Abstract: In this paper we obtain bounds for the order and exponent of the Schur multiplier of a p-group of given coclass. These are further improved for p-groups of maximal class. In particular, we prove that if G is p-group of maximal class, then |H2(G, ℤ)| < |G| and expH2(G, ℤ) ≤ expG. The bound for the order can be improved asymptotically.

Remez-type inequality for discrete sets

Abstract: The classical Remez inequality bounds the maximum of the absolute value of a polynomial P(x) of degree d on [−1, 1] through the maximum of its absolute value on any subset Z of positive measure in [−1, 1]. Similarly, in several variables the maximum of the absolute value of a polynomial P(x) of degree d on the unit cube Q 1 ⊂ ℝ n can be bounded through the maximum of its absolute value on any subset Z ⊂ Q 1 of positive n-measure. The main result of this paper is that the n-measure in the Remez inequality can be replaced by a certain geometric invariant ω d (Z) which can be effectively estimated in terms of the metric entropy of Z and which may be nonzero for discrete and even finite sets Z.

Connected subgroups of SO(2,n) acting irreducibly on $\R^{2,n}$

Abstract: We classify all connected subgroups of SO(2, n) that act irreducibly on ℝ2, n. Apart from SO0(2, n) itself these are U(1, n/2), SU(1, n/2), if n even, S1 · SO(1, n/2) if n even and n ≥ 2, and SO0(1, 2) for n = 3. Our proof is based on the Karpelevich Theorem and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian conformal structures, namely SO0(2, n), SU(1, n), and SO0(1, 2).

On the abelian fundamental group scheme of a family of varieties

Abstract: Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π1(X, x)ab → π1(AlbX/S, \({0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}\)) induced by the canonical morphism from X to its Albanese scheme AlbX/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NSX/Sτ)⋎ → π1(X, x)ab → π1(AlbX/S, \({0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}\)) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber Xη of X can be extended to a finite and commutative pointed torsor over X.

Finite-dimensional absolute-valued algebras ∗

Abstract: In this paper we provide some new tools for the study of finite-dimensional absolute-valued algebras. We introduce homotopy notions in this field and develop some of their applications. Next, we parametrize these algebras by spin groups and study their isomorphisms. Finally, we introduce a duplication process for the construction of absolute-valued algebras.

MODEL-THEORETIC INDEPENDENCE IN THE BANACH LATTICES Lp(µ)

Abstract: We study model-theoretic stability and independence in Banach lattices of the form L p (X, U, µ), where 1 ≤ p < ∞. We characterize non-dividing using concepts from analysis and show that canonical bases exist as tuples of real elements.