Showing papers in "Israel Journal of Mathematics in 2005"

Limit groups as limits of free groups

Abstract: We give a topological framework for the study of Sela'slimit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known to coincide with the class of finitely generated fully residually free groups. The topological approach gives some new insight on the relation between fully residually free groups, the universal theory of free groups, ultraproducts and non-standard free groups.

Sum-free sets in abelian groups

Abstract: LetA be a subset of an abelian groupG with |G|=n. We say thatA is sum-free if there do not existx, y, z eA withx+y=z. We determine, for anyG, the maximal densityμ(G) of a sum-free subset ofG. This was previously known only for certainG. We prove that the number of sum-free subsets ofG is 2(μ(G)+o(1))n, which is tight up to theo-term. For certain groups, those with a small prime factor of the form 3k+2, we are able to give an asymptotic formula for the number of sum-free subsets ofG. This extends a result of Lev, Luczak and Schoen who found such a formula in the casen even.

Convergence of multiple ergodic averages along polynomials of several variables

Abstract: LetT be an invertible measure preserving transformation of a probability measure spaceX. Generalizing a recent result of Host and Kra, we prove that the averages $$T^{p1(u)} f$$ converge inL 2 (X) for anyf 1 ,…,f r ∃L ∞ (X), any polynomialsp 1 ,…,p r :f 1 ,...,:f 1 ∈1\t8(X and and Folner sequence \s{\gF r \s} \t8 in ℤ d .

Convergence of polynomial ergodic averages

Abstract: We prove theL2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials.

Ramanujan complexes of typeà d

Abstract: We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the buildings of type Ad-1 associated with PGLd(F) where F is a local field of positive characteristic.

Extending partial isometries

Abstract: We show that a finite metric spaceA admits an extension to a finite metric spaceB so that each partial isometry ofA extends to an isometry ofB. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space.

Badly approximable vectors on fractals

Abstract: For a large class of closed subsetsC of ℝn, we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.

Expanders, rank and graphs of groups

Abstract: Let G be a finitely presented group, and let (Gi} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. Gi is an amalgamated free product or HNN extension, for infinitely many i; 2. the Cayley graphs of G/Gi (with respect to a fixed finite set of generators for G) form an expanding family; 3. infi(d(Gi) - 1)/[G : Gi] -- O, where d(Gi) is the rank of Gi. The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.

Compositions of Random Transpositions

Abstract: LetY=(y1,y2, ...),y1≥y2≥..., be the list of sizes of the cycles in the composition ofcn transpositions on the set {1, 2, ...,n}. We prove that ifc>1/2 is constant andn → ∞, the distribution off(c)Y/n converges toPD(1), the Poisson-Dirichlet distribution with parameter 1, where the functionf is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that thePD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process.

Random complex zeroes, III. Decay of the hole probability

Abstract: The ‘hoe probability’ that a random entire function $$\psi (z) = \sum\limits_{k = 0}^\infty {\zeta _k \frac{{z^k }}{{\sqrt {k!} }}} ,$$ where ζ0, ζ1, ... are Gaussian i.i.d. random variables, has no zeroes in the disc of radiusr decays as exp(−cr 4) for larger.

Polynomial averages converge to the product of integrals

Abstract: We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL2 to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds.

Genericity of simple eigenvalues for a metric graph

Abstract: We prove that, for a metric graph different from a polygon, the spectrum of the Laplacian is generically simple.

The automorphism group of the Gaussian measure cannot act pointwise

Abstract: Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for BooleanG-actions. We show that in full generality this theorem does not hold for actions of Polish groups. In particular there is no Borel model for the Polish automorphism group of a Gaussian measure. In fact, we show that this group as well as many other Polish groups do not admit any nontrivial Borel measure preserving actions.

Diophantine geometry over groups III: Rigid and solid solutions

Abstract: This paper is the third 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 third paper we analyze exceptional families of solutions to a parametric system of equations. The structure of the exceptional solutions, and the global bound on the number of families of exceptional solutions we obtain, play an essential role in our approach towards quantifier elimination in the elementary theory of a free group presented in the next papers of this series. The argument used for proving the global bound is a key in proving the termination of the quantifier elimination procedure presented in the sixth paper of the series.

Series of independent random variables in rearrangement invariant spaces: An operator approach

Abstract: This paper studies series of independent random variables in rearrangement invariant spaces X on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(Lp), 1 ~ p ~ c~ and Lorentz spaces A¢. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumption Lp C X, p < oc is sufficient to guarantee that convergence of such series in X (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence in X of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result.

Un Théorème de Fatou Pour les Densités Conformes Avec Applications Aux Revêtements Galoisiens en Courbure Négative

Abstract: On etablit un theoreme analogue a celui de Fatou en theorie du potentiel, portant sur les densites conformes invariantes par certains groupes discrets d’isometries en courbure negative en place des fonctions harmoniques, ce qui permet d’obtenir de facon elementaire divers resultats concernant les revetements galoisiens en courbure negative, les uns nouveaux, les autres deja abordes autrement, comme l’ergodicite du feuilletage horospherique dans certains revetements nilpotents.

Bessel identities in the waldspurger correspondence over the real numbers

Abstract: We prove certain identities between Bessel functions attached to irreducible unitary representations of PGL2(R) and Bessel functions attached to irreducible unitary representations of the double cover of SL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. CONTENTS

On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

Abstract: In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL 2(0, 1).

Rigidity of measures -- the high entropy case and non-commuting foliations

Abstract: We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures.

Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism

Abstract: We study the existence of solutions to the orthogonal dynamics equation, which arises in the Mori-Zwanzig formalism in irreversible statistical mechanics. This equation generates the random noise associated with a reduction in the number of variables. IfL is the Liouvillian, or Lie derivative associated with a Hamiltonian system, andP an orthogonal projection onto a closed subspace ofL 2, then the orthogonal dynamics is generated by the operator (I −P)L. We prove the existence of classical solutions for the case whereP has finite-dimensional range. In the general case, we prove the existence of weak solutions.

On the Nitsche conjecture for harmonic mappings in ℝ2 and ℝ3

Abstract: We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r2+1).

Diophantine geometry over groups V1: Quantifier elimination I

Abstract: This paper is the first part (out of two) of the fifth paper in a sequence 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 two papers on quantifier elimination we use the iterative procedure that validates the correctness of anAE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra ofAE sets defined over a free group is invariant under projections, and hence show that every elementary set defined over a free group is in the Boolean algebra ofAE sets. The procedures we use for quantifier elimination, presented in this paper and its successor, enable us to answer affirmatively some of Tarski's questions on the elementary theory of a free group in the sixth paper of this sequence.

Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval

Abstract: We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. forC1+α expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.

Multiplicatively large sets and ergodic Ramsey theory

Abstract: Multiplicatively large sets are defined in (ℕ, ·) by an analogy to sets of positive upper density in (ℕ, +). By utilizing various ergodic multiple recurrence theorems, we show that multiplicatively large sets have a rich combinatorial structure. In particular, it is proved that for any multiplicatively large setE ⊂ ℕ and anyk ∈ ℕ, there existsa,b,c,d,e,q ∈ ℕ such that {fx23-1}

A classification for 2-isometries of noncommutative LP-spaces

Abstract: In this paper we extend previous results of Banach, Lamperti and Yeadon on isometries ofLp-spaces to the non-tracial case first introduced by Haagerup. Specifically, we use operator space techniques and an extrapolation argument to prove that every 2-isometryT:Lp(M) →Lp(N) between arbitrary noncommutativeLp-spaces can always be written in the form\(T(\phi ^{\frac{1}{p}} ) = w(\phi o \pi ^{ - 1} o {\rm E})^{\frac{1}{p}} , \phi \in \mathcal{M}_ * ^ + \) Here π is a normal *-isomorphism fromM onto the von Neumann subalgebra π(M) ofN,w is a partial isometry inN, andE is a normal conditional expectation fromN onto π(M). As a consequence of this, any 2-isometry is automatically a complete isometry and has completely contractively complemented range.

Any smooth plane quartic can be reconstructed from its bitangents

Abstract: In this paper we solve a classical reconstruction problem: can a smooth plane quartic be reconstructed from its bitangents? This was long known to be true if the symplectic structure on the bitangents is known. Our approach is to show, using the classification of 2-transitive groups, that the symplectric structure on the bitangents is unique.

Gaussian fluctuations for random walks in random mixing environments

Abstract: We consider a class of ballistic, multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions. Continuing our previous work [2] for the law of large numbers, we prove here that the fluctuations are Gaussian when the environment is Gibbsian satisfying the “strong mixing condition” of Dobrushin and Shlosman and the mixing rate is large enough to balance moments of some random times depending on the path. Under appropriate assumptions the annealed Central Limit Theorem (CLT) applies in both nonnestling and nestling cases, and trivially in the case of finite-dependent environments with “strong enough bias”. Our proof makes use of the asymptotic regeneration scheme introduced in [2]. When the environment is only weakly mixing, we can only prove that if the fluctuations are diffusive then they are necessarily Gaussian.

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

Abstract: We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ∪L p , based on the size of the norms of sums of sub-blocks of the firstn functions. The results are aplied to the study of a.e. convergence of series Σ n a n T n g/ n whenT is anL 2 -contraction,g∃L 2 , and {a n } is an appropriate sequence. Given a sequence {f n }∪L p (Ω, μ), 1

CCC forcing and splitting reals

Abstract: The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.

Galois embedding problems with cyclic quotient of orderp

Abstract: LetK/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable\(\mathbb{F}_p \left[ {Gal\left( {K/F} \right)} \right] - modules\). For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.