Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts

1997 ◽  
Vol 123 (2) ◽  
pp. 149-170 ◽  
Author(s):  
M. S. M. Noorani
Keyword(s):  
Weak Mixing ◽  
Markov Shifts ◽  
Measure Preserving Transformations

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 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.

Refined scales of weak-mixing dynamical systems: typical behaviour

2019 ◽  
Vol 40 (12) ◽  
pp. 3296-3309
Author(s):  
SILAS L. CARVALHO ◽  
CÉSAR R. DE OLIVEIRA
Keyword(s):  
Invariant Measures ◽  
Dynamical Behaviour ◽  
Lebesgue Spaces ◽  
Weak Mixing ◽  
Axiom A ◽  
Typical Behaviour ◽  
Topologically Mixing ◽  
Continuous Measures ◽  
Basic Sets ◽  
Measure Preserving Transformations

We study sets of measure-preserving transformations on Lebesgue spaces with continuous measures taking into account extreme scales of variations of weak mixing. It is shown that the generic dynamical behaviour depends on subsequences of time going to infinity. We also present corresponding generic sets of (probability) invariant measures with respect to topological shifts over finite alphabets and Axiom A diffeomorphisms over topologically mixing basic sets.

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 Ergodic and mixing sequences of transformations

1984 ◽  
Vol 4 (3) ◽  
pp. 353-366 ◽  
Author(s):  
Daniel Berend ◽  
Vitaly Bergelson
Keyword(s):  
Probability Space ◽  
Strong Mixing ◽  
Compact Groups ◽  
Ergodic Theorems ◽  
Affine Transformations ◽  
Weak Mixing ◽  
Mixing Sequences ◽  
Measure Preserving Transformations

AbstractThe notions of ergodicity, strong mixing and weak mixing are defined and studied for arbitrary sequences of measure-preserving transformations of a probability space. Several results, notably ones connected with mean ergodic theorems, are generalized from the case of the sequence of all powers of a single transformation to this case. The conditions for ergodicity, strong mixing and weak mixing of sequences of affine transformations of compact groups are investigated.

scholarly journals Conditions for rational weak mixing

2016 ◽  
Vol 16 (02) ◽  
pp. 1660004 ◽  
Author(s):  
Jon. Aaronson
Keyword(s):  
Weak Mixing ◽  
Weakly Mixing ◽  
Measure Preserving Transformations

We exhibit rationally ergodic, spectrally weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.

New proofs that weak mixing is generic

1976 ◽  
Vol 32 (3) ◽  
pp. 263-278 ◽  
Author(s):  
Steven Alpern
Keyword(s):  
Weak Mixing
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