Showing papers in "Israel Journal of Mathematics in 2007"

Hamiltonian dynamics on convex symplectic manifolds

Abstract: We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results.

On De Jong’s conjecture

Abstract: Let X be a smooth projective curve over a finite field F q . Let ρ be a continuous representation π(X) → GL n (F), where F = F l ((t)) with F l being another finite field of order prime to q. Assume that $$ \rho \left| {_{\pi (\bar X)} } \right. $$ is irreducible. De Jong’s conjecture says that in this case $$ \rho (\pi (\bar X)) $$ is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture. In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.

Automatic continuity of homomorphisms and fixed points on metric compacta

Abstract: We prove that arbitrary homomorphisms from one of the groups $$Homeo(2^\mathbb{N} ), Homeo(2^\mathbb{N} )^\mathbb{N} , Aut(\mathbb{Q}, < ), Homeo(\mathbb{R}) or Homeo(S^1 )$$ into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result of V. G. Pestov, that any action of the discrete group Homeo+(ℝ) by homeomorphisms on a compact metric space has a fixed point.

Small ball probability and Dvoretzky’s Theorem

Abstract: Large deviation estimates are by now a standard tool in Asymptotic Convex Geometry, contrary to small deviation results In this note we present a novel application of a small deviations inequality to a problem that is related to the diameters of random sections of high dimensional convex bodies Our results imply an unexpected distinction between the lower and upper inclusions in Dvoretzky’s Theorem

On the Fourier tails of bounded functions over the discrete cube

Abstract: In this paper we consider bounded real-valued functions over the discrete cube, f: {−1, 1}n → [−1, 1]. Such functions arise naturally in theoretical computer science, combinatorics, and the theory of social choice. It is often interesting to understand when these functions essentially depend on few coordinates. Our main result is a dichotomy that includes a lower bound on how fast the Fourier coefficients of such functions can decay: we show that $$\sum\limits_{|S| > k} {\hat f(S)^2 \geqslant exp( - O(k^2 logk))} ,$$ unless f depends essentially only on 2 O(k) coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor. p ]The same type of result has already been proven (and shown useful) for Boolean functions [Bou02, KS]. The proof of these results relies on the Booleanity of the functions, and does not generalize to all bounded functions. In this work we handle all bounded functions, at the price of a much faster tail decay. As already mentioned, this rate of decay is shown to be both roughly necessary and sufficient. p ]Our proof incorporates the use of the noise operator with a random noise rate and some extremal properties of the Chebyshev polynomials.

Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category

Abstract: If H is a Hopf algebra with bijective antipode and α, β ∈ Aut Hopf (H), we introduce a category $$_H \mathcal{Y}\mathcal{D}^H (\alpha ,\beta )$$ , generalizing both Yetter-Drinfeld modules and anti-Yetter-Drinfeld modules. We construct a braided T-category $$\mathcal{Y}\mathcal{D}(H)$$ having all the categories $$_H \mathcal{Y}\mathcal{D}^H (\alpha ,\beta )$$ as components, which, if H is finite dimensional, coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (α, β) admits a so-called pair in involution, then $$_H \mathcal{Y}\mathcal{D}^H (\alpha ,\beta )$$ is isomorphic to the category of usual Yetter-Drinfeld modules $$_H \mathcal{Y}\mathcal{D}^H $$ .

A local variational principle of pressure and its applications to equilibrium states

Abstract: We prove a local variational principle of pressure for any given open cover. More precisely, for a given dynamical system (X, T), an open cover $$\mathcal{U}$$ of X, and a continuous, real-valued function f on X, we show that the corresponding local pressure P(T, f; $$\mathcal{U}$$ ) satisfies $$P(T,f;\mathcal{U}) = \sup \left\{ {h_\mu (T,\mathcal{U}) + \int_X {f(x)\mu (x):\mu } is a T - invariant measure} \right\}$$ , moreover, the supremum can be attained by a T-invariant ergodic measure. By establishing the upper semi-continuity and affinity of the entropy map relative to an open cover, we further show that $$P(T,f;\mathcal{U}) = \sup \left\{ {h_\mu (T,\mathcal{U}) + \int_X {f(x)\mu (x):\mu } is a T - invariant measure} \right\}$$ for any T-invariant measure μ of (X, T), i.e., local pressures determine local measure-theoretic entropies. As applications, properties of both local and global equilibrium states for a continuous, real-valued function are studied.

Colored-descent representations of complex reflection groups G(r, p, n)

Abstract: We study the complex reflection groups G(r, p, n). By considering these groups as subgroups of the wreath products Open image in new window, and by using Clifford theory, we define combinatorial parameters and descent representations of G(r, p, n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.

Algebras in sets of queer functions

Abstract: We show that the set of continuous nowhere differentiable functions, the set of Dirichlet series which are bounded in the right half-plane and diverge everywhere on the imaginary axis, and the set of continuous interpolating functions contain big algebras.

Injective linear cellular automata and sofic groups

Abstract: Let V be a finite-dimensional vector space over a field \(\mathbb{K}\) and let G be a sofic group. We show that every injective linear cellular automaton τ: VG → VG is surjective. As an application, we obtain a new proof of the stable finiteness of group rings of sofic groups, a result previously established by G. Elek and A. Szabo using different methods.

On symmetric finsler spaces

Abstract: In this paper, we study symmetric Finsler spaces. We first study some geometric properties of globally symmetric Finsler spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Then we prove that a globally symmetric Finsler space is a Berwald space. As an application, we use the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Finsler spaces and obtain the formulas for flag curvature and Ricci scalar. Finally, some rigidity results of locally symmetric Finsler spaces related to the flag curvature are also given.

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Abstract: Let F k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F k → Aut(X) corresponds to a free action of F k on a simplicial tree X, in particular, where φ corresponds to the action of F k on its Cayley graph via an automorphism of F k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F k ). We also prove that the set of all λ(φ), where φ varies over Aut(F k ), has a gap between 1 and 1+(2k−3)/(2k 2−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ).

On maps between modular Jacobians and Jacobians of Shimura curves

Abstract: Fix a squarefree integer N, divisible by an even number of primes, and let Γ′ be a congruence subgroup of level M, where M is prime to N For each D dividing N and divisible by an even number of primes, the Shimura curve XD(Γ0(N/D) ∩ Γ′) associated to the indefinite quaternion algebra of discriminant D and Γ0(N/D) ∩ Γ′-level structure is well defined, and we can consider its Jacobian JD(Γ0(N/D) ∩ Γ′) Let JD denote the N/D-new subvariety of this Jacobian

On the positivity set of a linear recurrence sequence

Abstract: We consider real sequences (f n ) that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive dominating characteristic root, we establish that the density is positive. Furthermore, we determine the values that can occur as density of such a positivity set, both for the special case just mentioned and in general.

Non-absolutely continuous foliations

Abstract: We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also establish a more general version of this result for general foliations with compact leaves.

Quasi-Lipschitz equivalence of fractals

Abstract: The paper proves that if E and F are dust-like C 1 self-conformal sets with $$0 < \mathcal{H}^{\dim _H E} (E),\mathcal{H}^{\dim _H F} (F) < \infty $$ , then there exists a bijection f: E a F such that $$\frac{{(dim_H F)log|f(x) - f(y)|}}{{(dim_H E)log|x - y|}} \to 1$$ uniformly as |x−y} a 0. It is also proved that a self-similar arc is Holder equivalent to [0, 1] if and only if it is a quasi-arc.

Automorphisms of the Hatcher-Thurston complex

Abstract: Let S be a compact, connected, orientable surface of positive genus. Let $$\mathcal{H}\mathcal{T}(S)$$ be the Hatcher-Thurston complex of S. We prove that Aut $$\mathcal{H}\mathcal{T}(S)$$ is isomorphic to the extended mapping class group of S modulo its center.

Extending lipschitz maps into c(k)-spaces

Abstract: We show that if K is a compact metric space then C(K) is a 2-absolute Lipschitz retract. We then study the best Lipschitz extension constants for maps into C(K) from a given metric space M, extending recent results of Lancien and Randrianantoanina. They showed that a finite-dimensional normed space which is polyhedral has the isometric extension property for C(K)-spaces; here we show that the same result holds for spaces with Gateaux smooth norm or of dimension two; a three-dimensional counterexample is also given. We also show that X is polyhedral if and only if every subset E of X has the universal isometric extension property for C(K)-spaces. We also answer a question of Naor on the extension of Holder continuous maps.

Invariant measures for the horocycle flow on periodic hyperbolic surfaces

Abstract: We classify the ergodic invariant Radon measures for the horocycle flow on geometrically infinite regular covers of compact hyperbolic surfaces. The method is to establish a bijection between these measures and the positive minimal eigenfunctions of the Laplacian of the surface. Two consequences arise: if the group of deck transformations G is of polynomial growth, then these measures are classified by the homomorphisms from G 0 to ℝ where G 0 ≤ G is a nilpotent subgroup of finite index; if the group is of exponential growth, then there may be more than one Radon measure which is invariant under the geodesic flow and the horocycle flow. We also treat regular covers of finite volume surfaces.

On measure concentration for separately Lipschitz functions in product spaces

Abstract: Let Mn = X1 + ⋯ + Xn be a martingale with bounded differences Xm = Mm − Mm−1 such that ℙ{am − σm ≤ Xm ≤ am + σm} = 1 with nonrandom nonnegative σm and σ(X1, …, Xm−1)-measurable random variables am. Write σ2 = σ12 + ⋯ + σn2. Let I(x) = 1 − Φ(x), where Φ is the standard normal distribution function. We prove the inequalities $$\mathbb{P}\left\{ {M_n \geqslant x} \right\} \leqslant cI(x/\sigma ), \mathbb{P}\left\{ {M_n > x} \right\} \geqslant 1 - cI( - x/\sigma )$$ with a constant c such that 3.74 … ≤ c ≤ 7.83 …. The result yields sharp bounds in some models related to the measure concentration. In the case where all am = 0 (or am ≤ 0), the bounds for constants improve to 3.17 … ≤ c ≤ 4.003 …. The inequalities are new even for independent X1, …, Xn, as well as for linear combinations of independent Rademacher random variables.

An Engel condition with generalized derivations on multilinear polynomials

Abstract: Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x 1,..., x n) a multilinear polynomial over C, I a nonzero right ideal of R. If [g(f(r 1,..., r n)), f(r 1,..., r n)] = 0, for all r 1, ..., r n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:

A conjecture of Ax and degenerations of Fano varieties

Abstract: James Ax conjectured that every pseudo algebraically closed field is C1. We prove this conjecture in characteristic 0 by relating it to degenerations of Fano varieties.

A Nichtnegativstellensatz for polynomials in noncommuting variables

Abstract: Let S⋃{f} be a set of symmetric polynomials in noncommuting variables. If f satisfies a polynomial identity Σihi*fhi = 1 + Σigi*sigi for some si ∈ S ⋃ {1}, then f is obviously nowhere negative semidefinite on the class of tuples of nonzero operators defined by the system of inequalities s ≥ 0 (s ∈ S). We prove the converse under the additional assumption that the quadratic module generated by S is Archimedean.

Defining equations for cyclic prime covers of the Riemann sphere

Abstract: We determine a method to find explicit defining equations for each compact Riemann surface which admits a cyclic group of automorphisms C p of prime order p such that the quotient space has genus 0.

Riesz transform and g-function associated with Bessel operators and their appropriate Banach spaces

Abstract: We study g-functions and Riesz transforms related to the Bessel operators $$ \Delta _\mu = - x^{\_\mu \_1/2} Dx^{2\mu + 1} Dx^{\_\mu \_1/2} . $$ The method we use allows us to characterize the Banach spaces \( \mathbb{B} \) for which these operators are bounded when acting on \( \mathbb{B} \)-valued functions.

Random weighting, asymptotic counting, and inverse isoperimetry

Abstract: For a family X of k-subsets of the set {1, …, n}, let |X| be the cardinality of X and let Γ(X, μ) be the expected maximum weight of a subset from X when the weights of 1, …, n are chosen independently at random from a symmetric probability distribution μ on ℝ. We consider the inverse isoperimetric problem of finding μ for which Γ(X, μ) gives the best estimate of ln |X|. We prove that the optimal choice of μ is the logistic distribution, in which case Γ(X, μ) provides an asymptotically tight estimate of ln |X| as k −1 ln |X} grows. Since in many important cases Γ(X, μ) can be easily computed, we obtain computationally efficient approximation algorithms for a variety of counting problems. Given μ, we describe families X of a given cardinality with the minimum value of Γ(X, μ), thus extending and sharpening various isoperimetric inequalities in the Boolean cube.

Matrix algebras of polynomial codimension growth

Abstract: We study associative algebras with unity of polynomial codimension growth. For any fixed degree k we construct associative algebras whose codimension sequence has the largest and the smallest possible polynomial growth of degree k. We also explicitly describe the identities and the exponential generating functions of these algebras.

Spaces of functions with countably many discontinuities

Abstract: Let Γ be a Polish space and let K be a separable and pointwise compact set of functions on Γ Assume further that each function in K has only countably many discontinuities It is proved that \(\mathcal{C}(K)\) admits a \(\mathfrak{T}_p \)-lower semicontinuous and locally uniformly rotund norm, equivalent to the supremum norm A slightly more general result is shown and a related conjecture is stated

Relative entropy tuples, relative u.p.e. and c.p.e. extensions

Abstract: Relative entropy tuples both in topological and measure-theoretical settings, relative uniformly positive entropy (rel.-u.p.e.) and relative completely positive entropy (rel.-c.p.e.) are studied. It is shown that a relative topological Pinsker factor can be deduced by the smallest closed invariant equivalence relation containing the set of relative entropy pairs. A relative disjointness theorem involving relative topological entropy is proved. Moreover, it is shown that the product of finite rel.-c.p.e. extensions is also rel.-c.p.e..

Pair correlations of sequences in higher dimensions

Abstract: We consider a system of “generalised linear forms” defined on a subset x = (x ij ) of ℝd by $$ L_1 (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k) = \sum\limits_{j = 1}^{d_1 } {g_{1j}^k (x_{1j} ), \ldots ,} L_l (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k) = \sum\limits_{j = 1}^{d_l } {g_{lj}^k (x_{lj} ) \in \mathbb{R}} , for k \geqslant 1, $$ , where d = d 1 + ⋯+ d l and for each pair of integers (i, j), 1 ≤ i ≤ l, 1 ≤ j ≤ d i the sequence of functions (g ij k (x)) =1 ∞ is differentiable on an interval X ij . Then let $$ X_k (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) = (\{ L_1 (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k)\} , \ldots ,\{ L_l (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )(k)\} ) \in \mathbb{T}^l , $$ , for x in the Cartesian product X = × =1 l × =1 di X ij ⊂ ℝd. Let R = I 1 × ⋯ × I l be a rectangle in $$ \mathbb{T}^l $$ and for each N ≥ 1 let $$ V_N (R) = \sum\limits_{1 \leqslant n e m \leqslant N} {\chi _R (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} _n (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{X} _m (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} ))} $$ and then define $$ \Delta _N = \mathop {\sup }\limits_{R \subset \mathbb{T}^l } \{ V_N (R) - N(N - 1) leb(R)\} $$ where the supremum is over all rectangles in $$ \mathbb{T}^l $$ . We show that for almost every x ∈ $$ \mathbb{T}^d $$ we have that $$ \Delta _N = O(N(\log N)^\alpha ), $$ for appropriate α. Other related results are also described.