scholarly journals Rigidity, weak mixing, and recurrence in abelian groups

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ethan M. Ackelsberg
Keyword(s):  
Abelian Groups ◽  
Ergodic Measure ◽  
Weak Mixing ◽  
Weakly Mixing ◽  
Torsion Groups ◽  
New Phenomena ◽  
Measure Preserving Actions

<p style='text-indent:20px;'>The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions extend to this setting:</p><p style='text-indent:20px;'>1. If <inline-formula><tex-math id="M2">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.</p><p style='text-indent:20px;'>2. There exists a sequence <inline-formula><tex-math id="M3">\begin{document}$ (r_n) $\end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.</p><p style='text-indent:20px;'>The first of these results was shown for <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type="bibr" rid="b1">1</xref>], Fayad and Thouvenot [<xref ref-type="bibr" rid="b20">20</xref>], and Badea and Grivaux [<xref ref-type="bibr" rid="b2">2</xref>]. The latter was established in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type="bibr" rid="b23">23</xref>]. While techniques for handling <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.</p><p style='text-indent:20px;'>As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>, while others exhibit new phenomena.</p>

scholarly journals Weak mixing for locally compact quantum groups

2016 ◽  
Vol 37 (5) ◽  
pp. 1657-1680 ◽  
Author(s):  
AMI VISELTER
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Discrete Groups ◽  
Weak Mixing ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Weakly Mixing

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

scholarly journals Partitions with independent iterates in random dynamical systems

2009 ◽  
Vol 30 (2) ◽  
pp. 361-377 ◽  
Author(s):  
BORIS BEGUN ◽  
ANDRÉS DEL JUNCO
Keyword(s):  
Dynamical Systems ◽  
Random Dynamical Systems ◽  
Weakly Mixing ◽  
Dense Set ◽  
Measure Preserving Actions

AbstractKrengel characterized weakly mixing actions (X,T) as those measure-preserving actions having a dense set of partitions of X with infinitely many jointly independent images under iterates of T. Using the tools developed in later papers—one by del Junco, Reinhold and Weiss, another by del Junco and Begun—we prove analogues of these results for weakly mixing random dynamical systems (in other words, relatively weakly mixing systems).

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

On Pure-High Subgroups of Abelian Groups

1974 ◽  
Vol 17 (4) ◽  
pp. 479-482 ◽  
Author(s):  
K. Benabdallah
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Torsion Groups ◽  
Mixed Groups

Fuchs, in [3], problem 14, proposes the study of pure-high subgroups of an abelian group. In this paper we show that in abelian torsion groups, pure-high subgroups are also high. A natural problem arises, that of characterizing the pure-absolute summands. We show that this concept is the same as absolute summands in torsion groups, but that it is more general in mixed abelian groups. There is a definite connection between the existence of pure TV-high subgroups and the splitting of mixed groups. The notation is that of [3].

On 2-Symmetric Words and Verbal Subgroup of Metabelian Product of Abelian Groups

2006 ◽  
Vol 13 (03) ◽  
pp. 535-540
Author(s):  
Jiangmin Pan
Keyword(s):  
Free Group ◽  
Abelian Groups ◽  
Verbal Subgroup ◽  
Metabelian Product ◽  
Rank 2 ◽  
Torsion Groups

Let F be the free group of rank 2 with basis {x, y}, and G a metabelian product of some non-trivial abelian groups. If not all the factors of G are torsion groups, it is proved that the verbal subgroup of G in F equals F″. Moreover, all the 2-symmetric words of G are determined by using left Fox derivatives. In addition, we provide an example to illustrate that if all the factors of G are torsion groups, the above results need not be true.

scholarly journals Point transitivity, -transitivity and multi-minimality

2014 ◽  
Vol 35 (5) ◽  
pp. 1423-1442 ◽  
Author(s):  
ZHIJING CHEN ◽  
JIAN LI ◽  
JIE LÜ
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Topological Dynamical System ◽  
Property A ◽  
Weak Mixing ◽  
Furstenberg Family ◽  
Strongly Mixing ◽  
Transitive Point ◽  
Weakly Mixing

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

scholarly journals Ergodic multiplier properties

2014 ◽  
Vol 36 (3) ◽  
pp. 794-815 ◽  
Author(s):  
ADI GLÜCKSAM
Keyword(s):  
Irreducible Representation ◽  
Heisenberg Group ◽  
Weak Mixing ◽  
Locally Compact ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Weakly Mixing

In this article we will extend ‘the weak mixing theorem’ for certain locally compact Polish groups (Moore groups and minimally weakly mixing groups). In addition, we will show that the Gaussian action associated with the infinite-dimensional irreducible representation of the continuous Heisenberg group,$H_{3}(\mathbb{R})$, is weakly mixing but not mildly mixing.

Mixing on Sequences

1983 ◽  
Vol 35 (2) ◽  
pp. 339-352 ◽  
Author(s):  
Nathaniel A. Friedman
Keyword(s):  
Ergodic Measure ◽  
General Definition ◽  
Weak Mixing ◽  
Independent Sequence ◽  
Density Zero ◽  
Measure Preserving Transformation ◽  
Upper Density ◽  
Mixing Sequences ◽  
Definition Of ◽  
Single Sequence

Our aim is to study the mixing sequences of a weak mixing transformation. An ergodic measure preserving transformation is weak mixing if and only if for each pair of sets there exists a sequence of density one on which the transformation mixes the sets [9]. An unpublished result of S. Kakutani implies there actually exists a single sequence of density one on which the transformation is mixing for all sets (see Section 3). This result motivated the general définition of a transformation being mixing on a sequence, as well as mixing of higher order on a sequence. Given a weak mixing transformation, there exist sequences along which it is mixing of all degrees. In particular, this is the case for an eventually independent sequence [7].In Section 3 it will be shown that if T is weak mixing but not mixing, then a sequence on which T is two-mixing must have upper density zero.

scholarly journals Dynamics of Weakly Mixing Nonautonomous Systems

2019 ◽  
Vol 29 (09) ◽  
pp. 1950123 ◽  
Author(s):  
Mohammad Salman ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Sufficient Conditions ◽  
Unit Interval ◽  
Nonautonomous System ◽  
Probability Measures ◽  
Topological Transitivity ◽  
Weak Mixing ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical System ◽  
Weakly Mixing

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

scholarly journals ERGODIC PROPERTIES OF BOGOLIUBOV AUTOMORPHISMS IN FREE PROBABILITY

2010 ◽  
Vol 13 (03) ◽  
pp. 393-411 ◽  
Author(s):  
FRANCESCO FIDALEO ◽  
FARRUKH MUKHAMEDOV
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Free Probability ◽  
Classical System ◽  
Weak Mixing ◽  
Von Neumann ◽  
Commutation Relations ◽  
Ergodic Properties ◽  
Weakly Mixing ◽  
Fixed Point Algebra

We show that some C*-dynamical systems obtained by free Fock quantization of classical ones, enjoy ergodic properties much stronger than their boson or fermion analogous. Namely, if the classical dynamical system (X, T, μ) is ergodic but not weakly mixing, then the resulting free quantized system (𝔊, α) is uniquely ergodic (w.r.t. the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system (X, T, μ) which is weakly mixing but not mixing. In this case, the free quantized system is uniquely weak mixing but not uniquely mixing. Finally, a free quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing C*-dynamical systems whose Gelfand–Naimark–Segal representation associated to the unique invariant state generates a von Neumann factor of one of the following types: I∞, II1, IIIλwhere λ ∈ (0, 1]. The resulting scenario is then quite different from the classical one. In fact, if a classical system is uniquely mixing, it is conjugate to the trivial one consisting of a singleton. For the sake of completeness, the results listed above are extended to the q-Commutation Relations, provided [Formula: see text]. The last result has a self-contained meaning as we prove that the involved C*-dynamical systems based on the q-Commutation Relations are conjugate to the corresponding one arising from the free case (i.e. q = 0), at least if [Formula: see text].

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