Multiple recurrence of Markov shifts and other infinite measure preserving transformations

2000 ◽  
Vol 117 (1) ◽  
pp. 285-310 ◽  
Author(s):  
Jon Aaronson ◽  
Hitoshi Nakada
Keyword(s):  
Multiple Recurrence ◽  
Infinite Measure ◽  
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.

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.

Conjugates of Infinite Measure Preserving Transformations

1988 ◽  
Vol 40 (3) ◽  
pp. 742-749
Author(s):  
S. Alpern ◽  
J. R. Choksi ◽  
V. S. Prasad
Keyword(s):  
Conjugacy Class ◽  
Ergodic Theory ◽  
Lebesgue Space ◽  
Finite Measure ◽  
Infinite Measure ◽  
Ergodic Automorphism ◽  
Measurable Set ◽  
Image Position ◽  
Measure Preserving Transformations

In this paper we consider a question concerning the conjugacy class of an arbitrary ergodic automorphism σ of a sigma finite Lebesgue space (X, , μ) (i.e., a is a ju-preserving bimeasurable bijection of (X, , μ). Specifically we proveTHEOREM 1. Let τ, σ be any pair of ergodic automorphisms of an infinite sigma finite Lebesgue space (X, , μ). Let F be any measurable set such thatThen there is some conjugate σ' of σ such that σ'(x) = τ(x) for μ-almost every x in F.The requirement that F ∪ τF has a complement of infinite measure is, for example, satisfied when F has finite measure, and in that case, the theorem was proved by Choksi and Kakutani ([7], Theorem 6).Conjugacy theorems of this nature have proved to be very useful in proving approximation results in ergodic theory. These conjugacy results all assert the denseness of the conjugacy class of an ergodic (or antiperiodic) automorphism in various topologies and subspaces.

scholarly journals Multiple recurrence for non-commuting transformations along rationally independent polynomials

2013 ◽  
Vol 35 (2) ◽  
pp. 403-411 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
PAVEL ZORIN-KRANICH
Keyword(s):  
Nilpotent Group ◽  
Constant Term ◽  
Single Variable ◽  
Multiple Recurrence ◽  
Arbitrary Measure ◽  
Measure Preserving Transformations

AbstractWe prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+ {p}_{i} (n)$, with rationally independent ${p}_{i} $ with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.

Multiple recurrence and infinite measure preserving odometers

1998 ◽  
Vol 108 (1) ◽  
pp. 37-44 ◽  
Author(s):  
Stanley Eigen ◽  
Arshag Hajian ◽  
Kim Halverson
Keyword(s):  
Multiple Recurrence ◽  
Infinite Measure
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