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 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 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.

Piecewise isometries, uniform distribution and 3log 2−π2/8

2011 ◽  
Vol 32 (6) ◽  
pp. 1862-1888 ◽  
Author(s):  
YITWAH CHEUNG ◽  
AREK GOETZ ◽  
ANTHONY QUAS
Keyword(s):  
Phase Space ◽  
Uniform Distribution ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Full Measure ◽  
Infinite Measure ◽  
Important Family ◽  
Large Numbers ◽  
Piecewise Isometries ◽  
Measure Preserving Transformations

AbstractWe use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

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.

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