Homotopy invariants of mappings to the circle

Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and I _r (a): (X × T) _r → \mathbbZ \mathbb{Z} is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence I _r (a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of Z-valued functions on (X × T) ^r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

The bieri-strebel invariant and homological finiteness conditions for metabelian groups

A group Γ has type F P_n if a trivial ℤΓ-module ℤ has a projective resolution P:…P_n → … → P₁ → P₀ → ℤ in which ℤΓ-module P_n,…P₁, P₀ are finitely generated. Let the finitely generated group Γ be a split extension of the Abelian group M by an Abelian group Q, suppose M is torsion free, and assume Γ∈F P_m, m≥2. Then the invariant ∑ _c ^M is m-tame. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 194–218, March–April, 1997.     

Distinguishing labeling of the actions of almost simple groups

Suppose Γ is a group acting on a set X, written as (Γ,X). An r-labeling f: X→{1,2, ..., r} of X is called distinguishing for (Γ,X) if for all σ∈Γ,σ≠1, there exists an element x∈X such that f(x)≠f(x ^σ ). The distinguishing number d(Γ,X) of (Γ,X) is the minimum r for which there is a distinguishing r-labeling for (Γ,X). If Γ is the automorphism group of a graph G, then d(Γ,V (G)) is denoted by d(G), and is called the distinguishing number of the graph G. The distinguishing set of Γ-actions is defined to be D*(Γ)={d(Γ,X): Γ acts on X}, and the distinguishing set of Γ-graphs is defined to be D(Γ)={d(G): Aut(G)≅Γ}. This paper determines the distinguishing set of Γ-actions and the distinguishing set of Γ-graphs for almost simple groups Γ.     

On the existence of automorphisms with simple Lebesgue spectrum

It is shown that ifT is a measure preserving automorphism on a probability space (Ω,B, m) which admits a random variable X₀ with mean zero such that the stochastic sequence X₀ o T_n,n ε ℤ is orthonormal and spans L₀ ²(Ω,B,m), then for any integerk ≠ 0, the random variablesX o T^nk,n ε ℤ generateB modulom.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

Polaroid operators,SVEP and perturbed Browder,Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T ^*+A ^*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Recurrence in unipotent groups and ergodic nonabelian group extensions

LetT be a measure-preserving and ergodic transformation of a standard probability space (X,S, μ) and letf:X → SUT _d (ℝ) be a Borel map into the group of unipotent upper triangulard ×d matrices. We modify an argument in [12] to obtain a sufficient condition for the recurrence of the random walk defined byf, in terms of the asymptotic behaviour of the distributions of the suitably scaled mapsf(n,x)=(fT ⁿ⁻¹·fT ⁿ⁻²…fT·f). We give examples of recurrent cocycles with values in the continuous Heisenberg group H₁(ℝ)=SUT₃(ℝ), and we use a recurrent cocycle to construct an ergodic skew-product extension of an irrational rotation by the discrete Heisenberg group H₁(ℤ)=SUT₃(ℤ). The author was partially supported by the FWF research project P16004-MAT.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).     

A Decomposition Theorem for Compact-Valued Henstock Integral

We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.     

Algebraic Independence by Approximation Method (II)

Let f (x) be a continued fraction with elements a _n x, where coefficients a _n are positive algebraic numbers. Using the criterion of [l] for any nonzero real algebraic numbers α₁,...,α_s with distinct absolute values the algebraic independence of the values f(α₁), ..., f(α_s) is proved under certain assumption concerning only with a _n . For some transcendental numbers ξ the algebraic independence of values f(ξ^j)(j∈ℤ) is also established. Received March 27, 1998, Accepted September 28, 1998     

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

Applications of operator semigroups to Fourier analysis

Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case)T={T ⁿ /n=0,1,2,...} or (continuous case)T={T(t)/t=≥0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesàro limits (asn→∞) of |〈T ⁿ f,f〉|², |〈T (n)f,f〉|² are computed (withn∈ℤ⁺ orn∈ℤ⁺). Specializing the Hilbert space to beL ²(T,μ) (discrete case) orL ²(ℝ,μ) (continuous case) where μ is a Borel probability measure on the circle group or the line, the Cesàro limit of (asn→±∞, with,n∈ℤ orn∈ℝ) is obtained and interpreted. Extensions toT ^M , and ℝ ^M are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context. Partially supported by an NSF grant     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.