scholarly journals On rationally ergodic and rationally weakly mixing rank-one transformations

2014 ◽  
Vol 35 (4) ◽  
pp. 1141-1164 ◽  
Author(s):  
IRVING DAI ◽  
XAVIER GARCIA ◽  
TUDOR PĂDURARIU ◽  
CESAR E. SILVA
Keyword(s):  
Weak Mixing ◽  
Infinite Measure ◽  
Markov Shifts ◽  
Rank One ◽  
Measure Spaces ◽  
Metric Invariant ◽  
Weakly Mixing

AbstractWe study the notions of weak rational ergodicity and rational weak mixing as defined by J. Aaronson [Rational ergodicity and a metric invariant for Markov shifts.Israel J. Math. 27(2) (1977), 93–123; Rational weak mixing in infinite measure spaces.Ergod. Th. & Dynam. Sys.2012, to appear.http://arxiv.org/abs/1105.3541]. We prove that various families of infinite measure-preserving rank-one transformations possess or do not posses these properties, and consider their relation to other notions of mixing in infinite measure.

scholarly journals Rational weak mixing in infinite measure spaces

2012 ◽  
Vol 33 (6) ◽  
pp. 1611-1643 ◽  
Author(s):  
JON AARONSON
Keyword(s):  
Density Ratio ◽  
Weak Mixing ◽  
Limit Property ◽  
Infinite Measure ◽  
Markov Shifts ◽  
Measure Spaces ◽  
Ratio Limit ◽  
Mixing Property ◽  
Theoretic Version ◽  
Measure Preserving Transformations

AbstractRational weak mixing is a measure theoretic version of Krickeberg’s strong ratio mixing property for infinite measure preserving transformations. It requires ‘density’ ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey’s strong ratio limit property. The power, subsequence version of the property is generic.

scholarly journals On multiple recurrence and other properties of ‘nice’ infinite measure-preserving transformations

2016 ◽  
Vol 37 (5) ◽  
pp. 1345-1368 ◽  
Author(s):  
JON AARONSON ◽  
HITOSHI NAKADA
Keyword(s):  
Geodesic Flows ◽  
Multiple Recurrence ◽  
Weak Mixing ◽  
Interval Maps ◽  
Infinite Measure ◽  
Markov Shifts ◽  
Measure Preserving Transformations

We discuss multiple versions of rational ergodicity and rational weak mixing for ‘nice’ transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.

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 On mixing in infinite measure spaces

1968 ◽  
Vol 74 (6) ◽  
pp. 1150-1156 ◽  
Author(s):  
U. Krengel ◽  
L. Sucheston
Keyword(s):  
Infinite Measure ◽  
Measure Spaces

scholarly journals Slow entropy and differentiable models for infinite-measure preserving ℤk actions

2012 ◽  
Vol 32 (2) ◽  
pp. 653-674 ◽  
Author(s):  
MICHAEL HOCHMAN
Keyword(s):  
Compact Manifold ◽  
Borel Measure ◽  
Infinite Measure ◽  
Group Of Diffeomorphisms ◽  
Measure Spaces

AbstractWe define ‘slow’ entropy invariants for ℤd actions on infinite measure spaces, which measure growth of itineraries at subexponential scales. We use this notion to construct infinite-measure preserving ℤ2 actions which cannot be realized as a group of diffeomorphisms of a compact manifold preserving a Borel measure, in contrast to the situation for ℤ actions, where every infinite-measure preserving action can be realized in this way.

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].

A new characterization of the Haagerup property by actions on infinite measure spaces

2020 ◽  
pp. 1-20
Author(s):  
THIEBOUT DELABIE ◽  
PAUL JOLISSAINT ◽  
ALEXANDRE ZUMBRUNNEN
Keyword(s):  
Finite Measure ◽  
Invariant Mean ◽  
Locally Compact ◽  
Haagerup Property ◽  
Infinite Measure ◽  
Measurable Functions ◽  
Measure Spaces ◽  
Definition Of

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$ -finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.

The Converse of the Dominated Ergodic Theorem in Hurewicz Setting

1991 ◽  
Vol 34 (3) ◽  
pp. 405-411
Author(s):  
László I. Szabó
Keyword(s):  
Ergodic Theorem ◽  
Orlicz Spaces ◽  
Maximal Inequality ◽  
Infinite Measure ◽  
Measure Spaces

AbstractThe converse of the dominated ergodic theorem in infinite measure spaces is extended to non-singular transformations, i.e. transformations that only preserve the measure of null sets. An inverse weak maximal inequality is given and then applied to obtain related results in Orlicz spaces.

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