Showing papers in "Israel Journal of Mathematics in 2001"

Gibbs states on the symbolic space over an infinite alphabet

Abstract: We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Holder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Dynamical Borel-Cantelli lemmas for gibbs measures

Abstract: LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsA n ⊃ X and μ-almost every pointx∈X the inclusionT n x∈A n holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.

An operator approach to zero-sum repeated games

Abstract: We consider two person zero-sum stochastic games. The recursive formula for the valuesvλ (resp.v n) of the discounted (resp. finitely repeated) version can be written in terms of a single basic operator Φ(α,f) where α is the weight on the present payoff andf the future payoff. We give sufficient conditions in terms of Φ(α,f) and its derivative at 0 for limv n and limvλ to exist and to be equal. We apply these results to obtain such convergence properties for absorbing games with compact action spaces and incomplete information games.

Thermodynamic formalism for null recurrent potentials

Abstract: We extend Ruelle’s Perron-Frobenius theorem to the case of Holder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(ϕ) denote the Gurevic pressure of ϕ and letL ϕ be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatL ϕ * ν=e P(ϕ)ν,L ϕ h=e P(ϕ) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL ϕ , and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.

Fractional Poisson equations and ergodic theorems for fractional coboundaries

Abstract: For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT α=Σ =1 ∞ a j T j , where {a j } are the coefficients in the power series expansion (1-t)α=1-Σ =1 ∞ a j t j in the open unit disk, which satisfya j >0 anda j >0 and Σ =1 ∞ a j =1. The operator calculus justifies the notation(I−T) α :=I−T α (e.g., (I−T 1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) α x=y, i.e.,y is a coboundary forT α . The fractional Poisson equation forT is the Poisson equation forT α . We show that if(I−T)X is not closed, then(I−T) α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ =1 ∞ T k y/k 1-α converges in norm, and conclude that lim n ‖(1/n 1-α)Σ =1 T k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL 1 of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) α L 1 for some 0<α<1, then the one-sided Hilbert transform Σ =1 ∞ T k f/k converges a.e. For 11−1/p=1/q, then Σ =1 ∞ T k f/k 1/p converges a.e., and thus (1/n 1/p ) Σ =1 T k f converges a.e. to zero. Whenf∈(I−T) 1/q L p (the case α=1/q), we prove that (1/n 1/p (logn)1/q )Σ =1 T k f converges a.e. to zero.

Categoricity of an abstract elementary class in two successive cardinals

Abstract: We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models, or existence of large cardinals). We prove (assuming a weak version of GCH around λ) that if ℜ is categorical in λ, λ+,LS(ℜ) ≤ λ and has intermediate number of models in λ++,then ℜ has a model in λ+++.

John's decompositions: selecting a large part

Abstract: We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositionsid = ∑x j ⊕x j , rather than on the coordinate one. The John's decomposition brings this result to the local theory of Banach spaces. As a consequence, we get a new lemma of Dvoretzky-Rogers type, where the contact points of the unit ball with its maximal volume ellipsoid play a crucial role. We then apply these results to embeddings ofl ∞ into finite dimensional spaces.

A conjecture on arithmetic fundamental groups

Abstract: The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth $$\overline {\mathbb{F}_\ell ((t))} $$ is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings.

On the group of automorphisms of a surfacexny=P(z)

Abstract: In this note the AK invariant of a surface in ℂ3 which is given byxny=P(z) wheren>1 and deg(P)=d>1 is computed. Then this information is used to find the group of automorphisms of this surface and the isomorphism classes of such surfaces.

Hasse invariants for Hilbert modular varieties

Abstract: Given a totally real fieldL of degreeg, we constructg Hasse invariants on Hilbert modular varieties in characteristicp and characterize their divisors. We show that these divisors give the type stratification defined by the action of\(\mathcal{O}_L \) on theαp-elementary subgroup. Under certain conditions, involving special values of zeta functions, the product of these Hasse invariants is the reduction of an Eisenstein series of weightp−1

Sum-free sets in abelian groups

Abstract: We show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is $$\left( {2^{ u (G)} - 1} \right)2^{\left| G \right|/2} + O\left( {2^{(1/2 - \delta )\left| G \right|} } \right)$$ whereν(G) is the number of even order components in the canonical decomposition ofG into a direct sum of its cyclic subgroups, and the implicit constant in theO-sign is absolute.

On the identities of the Grassmann algebras in characteristicp>0

Abstract: In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM 2(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM 2(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM 2(K).

Directional derivatives of Lipschitz functions

Abstract: Letf be a Lipschitz mapping of a separable Banach spaceX to a Banach spaceY. We observe that the set of points at whichf is differentiable in a spanning set of directions but not Gâteaux differentiable isσ-directionally porous. Since Borelσ-directionally porous sets, in addition to being first category sets, are null in Aronszajn’s (or, equivalently, in Gaussian) sense, we obtain an alternative proof of the infinite-dimensional generalisation of Rademacher’s Theorem (due to Aronszajn) on Gâteaux differentiability of Lipschitz mappings. Better understanding ofσ-directionally porous sets leads us to a new version of Rademacher’s theorem in infinite dimensional spaces which we show to be stronger then the one obtained by Aronszajn. A more detailed analysis shows that (a stronger version of) our observation follows from a somewhat technical result showing that the behaviour of the slopes (f(x+t (u+v))−f(x+tv))/t ast → 0+is in some sense independent ofv. In particular, this implies that in the case of Lipschitz real valued functions the upper one-sided derivatives coincide with the derivatives defined by Michel and Penot, except for points of aσ-directionally porous set. This has a number of interesting consequences for upper and lower directional derivatives. For example, for allx ∈ X, except those which belong to aσ-directionally porous set, the functionv → $$\bar f$$ (x, υ) (the upper right derivative off atx in the directionv) is convex.

Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche

Abstract: Letw=f(z) be a univalent harmonic mapping of the annulus {ρ≤|z|≤1} onto the annulus {σ≤|w|≤1}. It is shown thatσ≤1/(1+(ρ 2/2)(logρ)2).

Transversals of additive Latin squares

Abstract: LetA={a 1, …,a k} andB={b 1, …,b k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S ksuch that the sums α i +b i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤ p ) a and\(\mathbb{Z}_{p^a } \) in the casek

Funny rank-one weak mixing for nonsingular abelian actions

Abstract: We construct funny rank-one infinite measure preserving free actionsT of a countable Abelian groupG satisfying each of the following properties: (1)T g1×…×Tgk is ergodic for each finite sequenceg 1,…,g k ofG-elements of infinite order, (2)T×T is nonconservative, (3)T×T is nonergodic but allk-fold Cartesian products are conservative, and theL ∞-spectrum ofT is trivial, (4) for eachg of infinite order, allk-fold Cartesian products ofT g are ergodic, butT 2g×Tg is nonconservative. A topological version of this theorem holds. Moreover, given an AT-flowW, we construct nonsingularG-actionsT with similar properties and such that the associated flow ofT isW. Orbit theory is used in an essential way here.

On Lie derivations of Lie ideals of prime algebras

Abstract: LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let $$\bar {\mathcal{R}} = {\mathcal{R}}/{\mathcal{Z}}\left( {\mathcal{R}} \right)$$ be the factor Lie algebra and let δ: $$\delta :\bar {\mathcal{R}} \to \bar {\mathcal{R}}$$ be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt 14, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A c such that $$x^D + {\mathcal{C}} = \left( {x + {\mathcal{C}}} \right)^\delta \in \left( {\mathcal{R} + {\mathcal{C}}} \right)/{\mathcal{C}} = \bar {\mathcal{R}}$$ for allx∈R. Our result solves an old problem of Herstein.

Range of cube-indexed random walk

Abstract: For given finite, connected, bipartite graphG=(V,E) with distinguishedν0 ∈V, set {fx189-1} Our main result says there is a fixedb so that whenG is a Hamming cube ({0, 1}n with the usual adjacency), andf is chosen uniformly fromF, the probability thatf takes more thanb values is at most e−Ω(n). this settles in a very strong way a conjecture of I. Benjamini, O. Haggstrom and E. Mossel [2].

The structure of uniformly Gâteaux smooth Banach spaces

Abstract: It is shown that a Banach spaceX admits an equivalent uniformly Gâteaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact

A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups

Abstract: A nilpotent group is defined whose local zeta functions counting subgroups and normal subgroups depend on counting points modp on the elliptic curvey2=x3−x. This example answers negatively a question raised in the paper of F. J. Grunewald, D. Segal and G. C. Smith where these local zeta functions were first defined. They speculated that local zeta functions of nilpotent groups might be finitely uniform asp varies. A proof is given that counting points on the elliptic curvey2=x3−x are not finitely uniform, and hence the same is true for the zeta function of the associated nilpotent group. This example demonstrates that nilpotent groups have a rich arithmetic beyond the connection with quadratic forms.

Extended Jacobson density theorem for rings with derivations and automorphisms

Abstract: We introduce and studyM-outer derivations and automorphisms and prove a version of the Chevalley-Jacobson density theorem for rings with such derivations and automorphisms.

A dimension gap for continued fractions with independent digits

Abstract: Kinney and Pitcher (1966) determined the dimension of measures on [0, 1] which make the digits in the continued fraction expansion i.i.d. variables. From their formula it is not clear that these dimensions are less than 1, but this follows from the thermodynamic formalism for the Gauss map developed by Walters (1978). We prove that, in fact, these dimensions are bounded by 1−10−7. More generally, we considerf-expansions with a corresponding absolutely continuous measureμ under which the digits form a stationary process. Denote byE δ the set of reals where the asymptotic frequency of some digit in thef-expansion differs by at leastδ from the frequency prescribed byμ. ThenE δ has Hausdorff dimension less than 1 for anyδ>0.

Periods and liftings: FromG 2 toC 3

Abstract: In this paper we characterize the exceptional theta lifting fromG 2 toC 3 by means of the existence of the simple pole ats=1 of the spin L-function forGSp 3 and by means of the nonvanishing of certain period onGSp 3. Among other results, we also prove the lifting fromG 2 toC 3 is functorial at local unramified places.

Derived subgroups of fixed points

Abstract: LetA be an elementary abelianq-group acting on a finiteq′-groupG. We show that ifA has rank at least 3, then properties ofC G(a)′, 1 ≠a ∈A restrict the structure ofG′. In particular, we consider exponent, order, rank and number of generators.

Generic representations for the unitary group in three variables

Abstract: We show that for the quasi-split unitary group in three variables every tempered packet of cuspidal automorphic representations contains a globally generic representation.

Metric entropy of convex hulls

Abstract: LetT be a precompact subset of a Hilbert space. The metric entropy of the convex hull ofT is estimated in terms of the metric entropy ofT, when the latter is of order eℒ2. The estimate is best possible. Thus, it answers a question left open in [CKP].

Holonomy on poisson manifolds and the modular class

Abstract: The linear holonomy of a Poisson structure, introduced in the present paper, generalizes the linearized holonomy of a regular symplectic foliation. For singular Poisson structures the linear holonomy is defined for the lifts of tangential paths to the cotangent bundle. The linear holonomy is closely related to the modular class. Namely, the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field along such a lift. This assertion relies on the notion of the integral of a vector field along a cotangent path on a Poisson manifold, which is also introduced in the paper.

Constructive lower bounds for off-diagonal Ramsey numbers

Abstract: We describe an explicit construction whicy, for some fixed absolute positive constant e, produces, for every integers>1 and all sufficiently largem, a graph on at least $$m^\varepsilon \sqrt {\log s/\log \log s} $$ vertices containing neither a clique of sizes nor an independent set of sizem.

Malnormality is undecidable in hyperbolic groups

Abstract: In answer to a question of Myasnikov, we show that there exist hyperbolic groups for which there is no algorithm to decide which finitely generated subgroups are malnormal or quasiconvex.

On the failure of the co-hopf property for subgroups of word-hyperbolic groups

Abstract: We provide an example of a finitely generated subgroupH of a torsion-free word-hyperbolic groupG such thatH is one-ended, andH does not split over a cyclic group, andH is isomorphic to one of its proper subgroups.