Automorphisms that admit an approximation by periodic transformations

Summary We are concerned with a class of automorphisms which can be approximated by periodic transformations. We show that such an automorphism T is ergodic and that a relationship exists between the spectrum of T, the measure of the sets which are invariant under T, and the ergodicity of the powers of T. Finally, we exhibit conditions under which T is weakly mixing and not strongly mixing.This research was supported by NSF Grant No. GP-7490.     

Strongly Generated Banach Spaces and Measures of Noncompactness

To generalize the Hausdorff measure of noncompactness to other classes of bounded sets (like e. g. conditionally weakly compact or Asplund sets), we introduce Grothendieck classes. We deduce integral inequalities for quantities (called Grothendieck measures) related to these classes. As a by-product, we can answer a question concerning the measure of noncompactness for linear T : X → Y introduced in [14], and generalize a theorem about weak solutions of differential equations in Banach spaces.     

Measure preserving transformations similar to Markov shifts

Similarity, that is, the existence of joint common extensions, defines an interesting equivalence relation for infinite measure preserving transformations T. We provide a sufficient condition, given in terms of return processes to reference sets of finite measure, for T to be similar to a Markov shift. This is then shown to apply to various piecewise smooth dynamical systems, including weakly hyperbolic transformations with indifferent periodic points or flat critical points, and discrete random walks driven by (weakly) hyperbolic maps.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

Subsequence generators for ergodic group translations

LetT be an ergodic translation on a compact abelian group. For every infinite set of integers {n _i} and ε >0 there is a setA of measure less than ε such that {T ⁿ _iA} generates the σ-algebra of measurable sets. Research partially supported by National Science Foundation Grants MCS7703659 (first author) and MCS7606735A01 (second author).     

Generic Stability and Essential Components of Generalized KKM Points and Applications

We propose the definition of T-KKM points and consider generic stability of T-KKM mappings and essential components of sets of T-KKM points. As applications, using a unified approach, we derive from these results the existence of essential components of solution sets to various optimization-related problems. We do this in two steps. First, we deduce the corresponding results for variational inclusions, which are new. Then we obtain, as consequences, the existence of essential components of solutions to other problems, which are new or include recent ones in the literature.     

Infinite measure preserving transformations with compact first regeneration

We study ergodic infinite measure preserving transformations T possessing reference sets of finite measure for which the set of densities of the conditional distributions given a first return (or entrance) at time n is precompact in a suitable function space. Assuming regular variation of wandering rates, we establish versions of the Darling-Kac theorem and the arcsine laws for waiting times and for occupation times which apply to transformations with indifferent orbits and to random walks driven by Gibbs-Markov maps. This research was supported by an APART [Austrian programme for advanced research and technology] fellowship of the Austrian Academy of Sciences. Much of this work was done at the Mathematics Department of Imperial College London. I also benefitted from a JRF at the ESI in Vienna.     

On Idomatic Partitions of Direct Products of Complete Graphs

Independent dominating sets in the direct product of four complete graphs are considered. Possible types of such sets are classified. The sets in which every pair of vertices agree in exactly one coordinate, called T ₁-sets, are explicitly described. It is proved that the direct product of four complete graphs admits an idomatic partition into T ₁-sets if and only if each factor has at least three vertices and the orders of at least two factors are divisible by 3.     

Finite groups acting 2-transitively on their sylow subgroups

In this paper we have completely determined: (1) all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups. (2) all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T. (3) all finite groups which are 2-transitive on all their sets of Sylow subgroups. The first author acknowledges the support of OPR Scholarship of Australia The second author is supported by the National Natural Science Foundation of China. Thanks are also due to the Department of Mathematics, the University of Western Australia, where he did his part of this work for its hospitality     

Residual behavior of induced maps

Consider (X,F, μ,T) a Lebesgue probability space and measure preserving invertible map. We call this a dynamical system. For a subsetA ∈F. byT _A:A →A we mean the induced map,T _A(x)=T^rA(x)(x) wherer _A(x)=min{i〉0:T ⁱ(x) ∈A}. Such induced maps can be topologized by the natural metricD(A, A’) = μ(AΔA’) onF mod sets of measure zero. We discuss here ergodic properties ofT _A which are residual in this metric. The first theorem is due to Conze.Theorem 1 (Conze):For T ergodic, T _A is weakly mixing for a residual set of A.Theorem 2:For T ergodic, 0-entropy and loosely Bernoulli, T _A is rank-1, and rigid for a residual set of A.Theorem 3:For T ergodic, positive entropy and loosely Bernoulli, T _A is Bernoulli for a residual set of A.Theorem 4:For T ergodic of positive entropy, T _A is a K-automorphism for a residual set of A. A strengthening of Theorem 1 asserts thatA can be chosen to lie inside a given factor algebra ofT. We also discuss even Kakutani equivalence analogues of Theorems 1–4.     

Stopping distributions for right processes

Summary LetX be a transient right process for which semipolar sets are polar. We characterize the measures which can arise as the distribution ofX _T withT a non-randomized stopping time.This work was done while the first-named author was visiting the University of California, San DiegoThe second-named author's research is supported in part by NSF grant DMS8721347     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

p-Helson sets, 1<p<2

Ap-Helson set is defined to be a closed subsetE of the circle groupT with the property that every continuous function onE can be extended to the full circle in such a way that this extension has its sequence of Fourier coefficients inl ^p. For 1<p<2, the union of two such sets is again ap-Helson set. It is shown that thep-Helson sets (p>1) differ from the Helson sets and also that the notion really depends on the indexp. An analogue of H. Helson’s result is given: ap-Helson set supports no nonzero measure with Fourier-Stieltjes transform inl ^q, 1/p+1/q=1.     

Weighted sums of squares in local rings and their completions, II

Let A be an excellent regular local ring of dimension two, let T be a finitely generated preordering in A, and let [^(T)]{\widehat T} be the preordering generated by T in the completion [^(A)]{\widehat A} of A. We study the question when the property of being saturated descends from [^(T)]{\widehat T} to T, and establish conditions of geometric nature which allow to decide this question. As an application we classify all principal preorderings in A of degree ≤ 3 which are saturated, in the case where A has a real closed residue field. These results have direct implications for nonnegativity certificates for real polynomials on two-dimensional semi-algebraic sets.     

Toppling kings in a tournament by introducing new kings

A king of a tournament is a vertex which can reach any other vertex within two steps. Let K be the king set of a tournament T. A subset M of K is c-topplable if there exists a supertournament T* of T such that there is a set N of c vertices in T * – T and (K – M) U N is exactly the king set of T*. A characterization of c-topplable sets of kings is given in this paper.     

Lipschitzian retracts and curves as invariant sets

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃ _i=1 ⁿ T _i (K) for some contractions T ₁, …, T _n of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.      

Ergodic automorphisms ofT n are Bernoulli shifts

An automorphism of then-dimensional torusT ⁿ, none of whose eigenvalues is a root of unity includes on the canonical measure space ofT ⁿ a measure preserving transformation which is isomorphic to a Bernoulli shift. This research was supported in part by the National Science Foundation grant GP-18884 and by the European Research Office of the U.S. Army contract DAJA-37-70-C-0701.     

NOTE – A Min-Max Relation for Monotone Path Systems in Simple Regions

xy -plane which bounds the simple polygonal (closed) region D. Let T and B be two finite, disjoint, equicardinal sets of points of D. We give a min-max relation for the maximum number of points of T and B which can be joined by a MPS in D, and a polytime algorithm for finding such a MPS. Received July 15, 1996/Revised October 7, 1999     

Two remarks on the relationship between BMO-regularity and analytic stability of interpolation for lattices of measurable functions

We study in this paper Hardy-type spaces on a measure space ( \mathbbT \mathbb{T} , m) × (Ω, μ), where ( \mathbbT \mathbb{T} , m) is the unit circle with Lebesgue measure. There is a characterization of analytic stability for real interpolation of weighted Hardy spaces on \mathbbT \mathbb{T} × Ω, a complete proof of which was present in the literature only for the case where μ is a point mass. Here this gap is filled, and a proof of the general case is presented. In a previous work by Kislyakov, certain results concerning BMO-regular lattices on ( \mathbbT \mathbb{T} × Ω, m × μ) were proved under the assumption that the measure μ is discrete. Here this extraneous assumption is lifted. Bibliography: 9 titles.     

A Decomposition of the Set of Subspaces Stable Under a Nilpotent Transformation

For a nilpotent linear transformation f: V → V of Jordan type λ, let S(T) be the set of f-stable subspaces W of V associated with an LR (Littlewood–Richardson) tableau T. We show that (i) S(T) is a disjoint union of locally closed sets parametrized by LR-fillings of T, each of which is birationally isomorphic to an affine space, and that (ii) a~conjecture of Shimomura on semistandard d-tableaux in a rectangular diagram is true.