Ergodic Group Rotations, Hartman Setsand Kronecker Sequences

 Let be a homomorphism with dense image in the compact group C. If is a continuity set, i.e. its topological boundary has Haar measure 0, then is called a Hartman set. If M is aperiodic then S contains the essential information about (C, ι) or, equivalently, about the dynamical system (C, T) where T is the ergodic group rotation . Using Pontryagin’s duality the paper presents a new method to get this information from S: The set S induces a filter on which is an isomorphism invariant for (C, T) and turns out to be a complete invariant for ergodic group rotations. If one takes , , , , one gets the interesting special case of Kronecker sequences (nα) which are classical objects in number theory and diophantine analysis.     

Closed Separator Sets

A smallest separator in a finite, simple, undirected graph G is a set S ⊆ V (G) such that G–S is disconnected and |S|=κ(G), where κ(G) denotes the connectivity of G. A set S of smallest separators in G is defined to be closed if for every pair S,T ∈ S, every component C of G–S, and every component S of G–T intersecting C either X(C,D) := (V (C) ∩ T) ∪ (T ∩ S) ∪ (S ∩ V (D)) is in S or |X(C,D)| > κ(G). This leads, canonically, to a closure system on the (closed) set of all smallest separators of G. A graph H with is defined to be S-augmenting if no member of S is a smallest separator in G ∪ H:=(V (G) ∪ V (H), E(G) ∪ E(H)). It is proved that if S is closed then every minimally S-augmenting graph is a forest, which generalizes a result of Jordán. Several applications are included, among them a generalization of a Theorem of Mader on disjoint fragments in critically k-connected graphs, a Theorem of Su on highly critically k-connected graphs, and an affirmative answer to a conjecture of Su on disjoint fragments in contraction critically k-connected graphs of maximal minimum degree.     

A nilpotent Roth theorem

Let T and S be invertible measure preserving transformations of a probability measure space (X, ℬ, μ). We prove that if the group generated by T and S is nilpotent, then exists in L ²-norm for any u, v∈L ^∞(X, ℬ, μ). We also show that for A∈ℬ with μ(A)>0 one has . By the way of contrast, we bring examples showing that if measure preserving transformations T, S generate a solvable group, then (i) the above limits do not have to exist; (ii) the double recurrence property fails, that is, for some A∈ℬ, μ(A)>0, one may have μ(A∩T _-n A∩S ^{- n} A)=0 for all n∈ℕ. Finally, we show that when T and S generate a nilpotent group of class ≤c, in L ²(X) for all u, v∈L ^∞(X) if and only if T×S is ergodic on X×X and the group generated by T ^-1 S, T ^-2 S ²,..., T ^-c S ^c acts ergodically on X. Oblatum 19-V-2000 & 5-VII-2001?Published online: 12 October 2001     

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.     

A simple map with no prime factors

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμ _S , i.e.,μ _S (A×B)=μ(A∩S ^;−n ;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕ _i=1 ^∞ ℤ₂, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.     

Strong orbit realization for minimal homeomorphisms

LetXbe a Cantor set,S a minimal self-homeomorphism ofX, and Μ anS-invariant Borel probability. LetT be an ergodic automorphism of a non-atomic Lebesgue probability space(Y,Ν). Then there is a minimal homeomorphismS′ with the same orbits asS such that (S′, Μ ) is measurably conjugate to (T, Ν). HereS′ can be chosen strongly orbit equivalent toS if and only if the periodic spectrum ofS is contained in the discrete spectrum ofT. Corollaries of these results generalize Dye’s Theorem and the Jewett-Krieger Theorem.     

Dye’s theorem in the almost continuous category

Suppose X and Y are Polish spaces with non-atomic Borel probability measures μ and ν and suppose that T and S are ergodic measure-preserving homeomorphisms of (X, μ) and (Y, ν). Then there are invariant G _δ subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a homeomorphism ϕ: X′ → Y′ which maps μ|X′ to ν|Y′ and maps T-orbits onto S-orbits. We also deal with the case where T and S preserve infinite invariant measures.     

On processes which cannot be distinguished by finite observation

A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions s _n such that s _n (x ₁,…, x _n ) → J(X) in probability for every process X=(x _n ) ∈ C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy [8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X and Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss’ result, and extends it to many other families of processes, e.g., it follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure. This research was supported by the Israel Science Foundation (grant No. 1333/04)     

A complex semigroup approach to group algebras of infinite dimensional Lie groups

A host algebra of a topological group G is a C ^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ^*-algebra C ^*(S,α), and we describe a setting which permits us to conclude that this C ^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future. To K.H. Hofmann on the occasion of his 75th birthday     

A local variational relation and applications

In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infh _μ(T,ℙ)≥h _top(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andh _μ(T,ℙ) > 0 for each ℙ finer thanu, then infh _μ(T,ℙ > 0 andh _top(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained. We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders). Project supported by one hundred talents plan and 973 plan.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

On F-almost split sequences

Let A be an Artinian algebra and F an additive subbifunctor of Ext,（-, -） having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod A and S a cotilting module over F = End（T）. Then Horn（-, T） induces a duality between F-almost split sequences in ⊥FT and almost sl31it sequences in ⊥S, where addrS = Hom∧（f（F）, T）. Let A be an F-Gorenstein algebra, T a strong F-cotilting module and 0→A→B→C→0 and F-almost split sequence in ⊥FT.If the injective dimension of S as a Г-module is equal to d, then C≌（ΩCM^-dΩ^dDTrA^＊）^＊,where（-）^＊=Hom（g,T）.In addition, if the F-injective dimension of A is equal to d, then A≌ΩMF^-dDΩFop^-d TrC≌ΩCMF^-d ≌F^d DTrC.     

On the Translates of Powers of a Continuous Periodic Function

We characterize the set of real-valued, 2π -periodic, continuous functions f for which the translation invariant subspace V(f) generated by f ⁿ , n≥0, is dense in C(\mathbbT)C(\mathbb{T}). In particular, it follows that if f takes a given value at only one point then V(f) is dense in C(\mathbbT)C(\mathbb{T}).     

A canonical structure theorem for finite joining-rank maps

A numerical isomorphism invariant,joining-rank, was introduced in [1] as a quantitative generalization of Rudolph’s property of minimal selfjoinings. Therein, a structure theory was developed for those transformationsT whose joining-rank, jr (T), is finite. Here, we sharpen the theorem and show it to be canonical: If jr (T)<∞ then there is a unique triple 〈e, p, S〉 wheree andp are natural numbers andS is a map with minimal self-joinings, such thatT is ane-point extension ofS ^P. Furthermore, the producte·p equals the joining-rank ofT. This theorem applies to any finite-rank mixing map, since for such maps the rank dominates the joining-rank. Another corollary is that any rank-1 transformation which is partial-mixing has minimal self-joinings. This partially answers a question of [3]. Partially supported by a National Science Foundation Postdoctoral Research Fellowship.     

Invariant measures and disintegrations with applications to Palm and related kernels

Consider a locally compact group G acting measurably on some spaces S and T. We prove a general representation of G-invariant measures on S and the existence of invariant disintegrations of jointly invariant measures on S × T. The results are applied to Palm and related kernels associated with a stationary random pair (ξ,η), where ξ is a random measure on S and η is a random element in T. An erratum to this article can be found at     

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.     

Unitary dilations and theC* algebra0 2

LetH be an infinite-dimensional separable Hilbert space, and letS=(S _ij)∈teB(H)⊗M ₂ be a unitary 2 × 2 matrix with operator entries. We study theC*-algebra generated by the operatorsS _ij, and show that the study of unitary dilations of isometriesT inH reduces to the special case whereS ₁₁ =T, andS ₂₁ = 0. We useC*-algebraic techniques to obtain detailed results about the set of all unitary dilations ofT. Work supported in part by NSF.     

Strictly ergodic models for non-invertible transformations

We generalize a result of R. Jewett [J]: IfT is an ergodic measure preserving transformation on (X, Ω,λ),T not necessarily invertible, there exists a strictly ergodicS acting on (Y, Θ,ν), whereY is compact, such that (X, Ω,λ, T) is measure theoretically isomorphic to (Y, Θ,ν, S).     

Colouring Arcwise Connected Sets in the Plane I

Let ? be the family of finite collections ? where ? is a collection of bounded, arcwise connected sets in ℝ² which for any S, T∈? where S∩T≠∅, it holds that S∩T is arcwise connected. We investigate the problem of bounding the chromatic number of the intersection graph G of a collection ?∈?.  Assuming G is triangle-free, suppose there exists a closed Jordan curve C⊂ℝ² such that C intersects all sets of ? and for all S∈?, the following holds: (i) S∩(C∪int (C)) is arcwise connected or S∩int (C)=∅. (ii) S∩(C∪ext (C)) is arcwise connected or S∩ext (C)=∅.  Here int(C) and ext (C) denote the regions in the interior, resp. exterior, of C. Such being the case, we shall show that χ(?) is bounded by a constant independent of ?. Revised: December 3, 1998     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.