Dispersing cocycles and mixing flows under functions

Let G G be a locally compact SIN group, and w w a weight on G G which is diagonally bounded with bound K K on a dispersed set. We show that the topological centre of L U C ( w − 1 ) ∗ {LUC}(w^{-1})^* equals M ( w ) M(w) , and there are ⌊ K ⌋ + 1 \lfloor K \rfloor + 1 many points in the spectrum of L U C ( w − 1 ) {LUC}(w^{-1}) which are determining for the topological centre (DTC). As corollaries, we obtain DTC results as well for L 1 ( w ) ∗ ∗ L_1(w)^{**} , C 0 ( w − 1 ) ⊥ C_0(w^{-1})^\perp and L ∞ , 0 ( w − 1 ) ⊥ L_{\infty , 0}(w^{-1})^\perp . We also give a short proof of the first result in the unweighted case for all locally compact second countable abelian groups G G .

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

To study outer actions $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0&lt;\lambda&lt;1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an AFD factor $\mathcal M$ of type $\mathrm{III}_{\lambda}$.

We make the following three observations regarding a question popularized by Katznelson: is every subset of \({\mathbb {Z}}\) which is a set of Bohr recurrence also a set of topological recurrence? (i) If G is a countable abelian group and \(E\subseteq G\) is an \(I_0\) set, then every subset of \(E-E\) which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of \(\{2^n-2^m : n,m\in {\mathbb {N}}\}\) which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let \({\mathbb {Z}}^{\omega }\) be the direct sum of countably many copies of \({\mathbb {Z}}\) with standard basis E. If every subset of \((E-E)-(E-E)\) which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime p and let \({\mathbb {F}}_p^\omega \) be the direct sum of countably many copies of \({\mathbb {Z}}/p{\mathbb {Z}}\) with basis \(({\mathbf {e}}_i)_{i\in {\mathbb {N}}}\). If for every p-uniform hypergraph with vertex set \({\mathbb {N}}\) and edge set \({\mathcal {F}}\) having infinite chromatic number, the Cayley graph on \(\mathbb F_p^\omega \) determined by \(\{\sum _{i\in F}{\mathbf {e}}_i:F\in {\mathcal {F}}\}\) has infinite chromatic number, then every subset of \(\mathbb F_p^\omega \) which is a set of Bohr recurrence is a set of topological recurrence.

Nous établissons que, pour tout groupe dénombrable abélien Γ et tout sous-groupe finiment engendré Δ de Γ, l’ensemble des actions de Δ sur un espace de probabilités standard (X,μ) qui peuvent être étendues en une action libre de Γ sur (X,μ) est générique (au sens de Baire). Ce résultat étend un théorème d’Ageev, qui correspond au cas où Δ est un groupe cyclique infini.

Sofic groups were defined implicitly by Gromov and explicitly by Weiss. All residually finite groups (and hence all linear groups) are sofic. The purpose of this paper is to introduce, for every countable sofic group G G , a family of measure-conjugacy invariants for measure-preserving G G -actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over G G , leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups, including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property (T) groups up to orbit equivalence and von Neumann equivalence, respectively.

A discrete norm on an Abelian groupA is a non-negative function ‖ · ‖A → ℝ which satisfies the triangle inequality, is homogenous with respect to scaling ofA by ℤ and is bounded away from 0 onA/{0}. A countable Abelian group is discretely normed if and only if the group is free.

Let K⊂ℝ d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in KΛ, equipped with the product topology, where each coordinate solves a SDE of the form dX i (t) = ∑ j a(j−i) (X j (t) −X i (t))dt + σ (X i (t))dB i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups Λ.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Kral’, Serra, and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct consequence of our removal result, we obtain the following generalization of Szemeredi’s theorem: for any compact abelian group G, any measurable set \({A \subseteq G}\) with Haar probability \({\mu(A) \geq \alpha > 0}\) satisfies \({\int_{G}\int_{G}1_{A}(x)1_{A}(x+r)...1_{A}(x+(k-1)r)\, {\rm d} \mu(x)\, {\rm d} \mu(r) \geq c}\), where the constant \({c=c(\alpha, k) > 0}\) is valid uniformly for all G. This result is shown to hold more generally for any translationinvariant system of r linear equations given by an integer matrix with coprime \({r \times r}\) minors.

We study homoclinic points of non-expansive automorphisms of compact abelian groups. Connections between the existence of non-trivial homoclinic points, expansiveness, entropy and adjoint automorphisms (in the sense of Einsiedler and Schmidt) are explored. Some implications for countable abelian group actions by automorphisms of compact abelian groups are also considered and it is shown that if every element of such an action has finite entropy, there can be no non-trivial common homoclinic points, unless the action is generated by a single automorphism.

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis double-struck upper Q comma greater-than right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Aut(\mathbb {Q},&gt;)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and, less trivially, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis script upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Aut(\mathcal {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the random graph. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.

We study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.