Maximal Abelian Subalgebras of the Hyperfinite Factor,Entropy and Ergodic Theory

Given a free ergodic action of a discrete abelian group G on a measure space (X, μ), the crossed product L ^∞ (X, μ)⋊ G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra. Received February 24, 2002, Accepted August 5, 2002     

Abelian varieties over cyclotomic fields with good reduction everywhere

 For every conductor f{1,3,4,5,7,8,9,11,12,15} there exist non-zero abelian varieties over the cyclotomic field Q(ζ _f ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X ₁ (f) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over Q(ζ _f ) with good reduction everywhere except possibly when f=11 or 15. Assuming the Generalized Riemann Hypothesis (GRH) we prove the same result when f=11 and 15. Received: 19 April 2001 / Revised version: 21 October 2001 / Published online: 10 February 2003     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle.     

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in ⁿ which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on ^s and ⁿ , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map and a constant C such that ||ϕ (x) - (x)|| for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | η; and if X is a Meyer set, then all algebraic conjugates | η ' | 1. Received May 9, 1997, and in revised form March 5, 1998.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">μ</Emphasis>) for vector-valued measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a measure from a σ-algebra of subsets of a set T into a sequentially complete Hausdorff topological vector space X. Assume that μ is convexly bounded, i.e., the convex hull of its range is bounded in X, and denote by L ¹(μ) the space of scalar valued functions on T which are integrable with respect to the vector measure μ. We study the inheritance of some properties from X to L ¹(μ). We show that the bounded multiplier property passes from X to L ¹(μ). Answering a 1972 problem of Erik Thomas, we show that for a rather large class of F-spaces X the non-containment of c ₀ passes from X to L ¹(μ).     

Convolution Operators on Banach Lattices with Shift-Invariant Norms

Let G be a locally compact abelian group and let μ be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57:105–111, 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol μ on such a space is bounded below by the L ^∞ norm of the Fourier–Stieltjes transform of μ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol μ is bounded above by the total variation of μ.     

The construction of a set of recurrence which is not a set of strong recurrence

This paper deals with two possible definitions of recurrence in measure preserving systems. A set of integersR is said to be a set of (Poincaré) recurrence if, for all measure preserving systems (X, B, μ, T) and any measurable setA of positive measure, there is anr εR such thatμ(T ^r A∩A)>0.R is said to be a set of strong recurrence if, for all measure preserving systems (X, B, μ, T) and any measurable setA of positive measure, there is ane>0 and an infinite number of elementsr ofR such thatμ(T ^r A∩A)≥e (see Bergelson’s 1985 paper). This paper constructs a set of recurrenceR, an example of a measure preserving system (X, B, μ, T) and a measurable setA of measure 1/2, such that lim _{r→∞:rεRμ} (A∩T ^r A)=0. In particularR is a set of recurrence but not a set of strong recurrence, giving a negative answer to a question of Bergelson posed in 1985. Further, it also constructs a set of recurrence which does not force the continuity of positive measures and so reproves a result of Bourgain published in 1987. This paper forms a part of the author’s Ph.D. Thesis at the Ohio State University. The author wishes to thank his advisor, Professor Bergelson, for suggesting the problem of this paper and for his guidance.     

Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)

Résumé. — Soit μ la mesure d'équilibre d'un endomorphisme de P ^k (C). Nous montrons ici qu'elle est son unique mesure d'entropie maximale. Nous construisons directement μ comme distribution asymptotique des préimages de tout point hors d'un ensemble exceptionnel algébraique.

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— Let μ be the equilibrium measure of an endomorphism of P ^k (C). We show that it is its unique measure of maximal entropy. We build μ directly as the distribution of premiages of any point outside an algebraic exceptional set.

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Manucsrit re?u le 30 novembre 2000.     

MAXIMAL CALDERON-ZYGMUND SINGULAR INTEGRAL ON RBMO

Given a positive Radon measure μ on R^d satisfying the linear growth condition μ（B（x,r））≤C0r^n,x∈R^d,r〉0,（1） where n is a fixed number and O〈n≤d. When d-1〈n,it is proved that if Tt,N1=0,then the corresponding maximal Calderon-Zygmund singular integral is bounded from RBMO to itself only except that it is infinite μ-a. e. on R^d.     

Specht modules with abelian vertices

In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p ²-cores where p is the characteristic of the underlying field. Furthermore, in the case of p≥3, or p=2 and μ is 2-regular, we show that the complexity of the Specht module S ^μ is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S^(pp)S^{(p^{p})} for p≥3.     

Splitting kernels into small summands

Let λ be a regular cardinal. An epimorphism between abelian groups is λ -pure if it is projective with respect to abelian groups of size less than λ. We show that cotorsion groups A have λ-pure projective dimension greater than 1 for all uncountable λ ≤ |A/tA|, where tA denotes the torsion subgroup of A. For λ > |A/tA|, cotorsion groups A are λ-pure projective.     

A Joint Integral Test for the Locations of Extrema for Brownian Motion

Let μ⁺(t) and μ⁻(t) be the locations of the maximum and minimum, respectively, of a standard Brownian motion in the interval [0,t]. We establish a joint integral test for the lower functions of μ⁺(t) and μ⁻(t), in the sense of Paul Lévy. In particular, it yields the law of the iterated logarithm for max(μ⁺(t),μ⁻(t)) as a straightforward consequence. Our result is in agreement with well-known theorems of Chung and Erdős [(1952) Trans. Amer. Math. Soc. 72, 179–186.], and Csáki, F?ldes and Révész [(1987) Prob. Theory Relat. Fields 76, 477–497].      

Congruences between derivatives of abelian <Emphasis Type="Italic">L</Emphasis>-functions at <Emphasis Type="Italic">s</Emphasis>=0

Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the ‘refined abelian Stark conjectures’ formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin–Stark conjecture and the ‘refined class number formulas’ of both Gross and Tate for all extensions K/k in which K is either an abelian extension of ℚ or is a function field. Mathematics Subject Classification (1991)  Primary 11G40; Secondary 11R65; 19A31; 19B28     

One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.     

Integrability and closest approximation representations

Summary Suppose U is a set,F is a field of subsets of U, andp _AB is the set of all real-valued bounded finitely additive functions defined onF. This paper consists of two main parts. In the first, a previously given (seeRiv. Math. Univ. Parma, (3),2 (1973), pp. 251–276) notion of a subset ofp _AB defined by certain closure properties and called a C-set, is considered, and those C-sets that are linear spaces are characterized. Now, suppose γ is a function whose domain isF and whose range is a collection of number sets with bounded union. The set,J _γ, of all elements ofp _AB with respect to which γ is integrable, for refinements of subdivisions, is a C-set and a linear space (seeRend. Sem. Mat. Univ. Padova,52 (1974), pp. 1–24). The second part of this paper concerns, for μ inp _AB and nonnegative-valued, a representation of the element ofJ _γ closest to μ with respect to variation norm. Entrata in Redazione il 14 giugno 1977.     

On Galois groups of unramified pro-p extensions

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _p -extensions of Q(μ _p ) and the Galois group of the maximal unramified pro-p extension of Q . We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and is in fact abelian.     

Random Matchings in Regular Graphs

d -regular graph G, let M be chosen uniformly at random from the set of all matchings of G, and for let be the probability that M does not cover x. We show that for large d, the 's and the mean μ and variance of are determined to within small tolerances just by d and (in the case of μ and ) : Theorem. For any d-regular graph G, (a) , so that , (b) , where the rates of convergence depend only on d. Received: April 12, 1996     

From free abelian groups to free abelian <Emphasis Type="Italic">ℓ</Emphasis>-groups

For each n, the free abelian group F on countably many free generators will be equipped with a monoid P _n making (F,P _n ) into the free n-generator abelian ℓ-group. This is a generalization of the main result of the second author, published in the J. Pure Appl. Algebra 208 (2007), 549–554, only dealing with the case n = 2.