scholarly journals Joint ergodicity along generalized linear functions

2015 ◽  
Vol 36 (7) ◽  
pp. 2044-2075 ◽  
Author(s):  
V. BERGELSON ◽  
A. LEIBMAN ◽  
Y. SON
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Linear Functions ◽  
Continuous Parameter ◽  
Greatest Integer Function ◽  
Similar Criterion ◽  
Probability Measure Space ◽  
Measure Preserving Transformations

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure-preserving transformations of $X$ and $\unicode[STIX]{x1D711}_{i}$ are integer-valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending on a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ along primes.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

On perturbed stochastic discrete systems

1974 ◽  
Vol 11 (3) ◽  
pp. 385-393 ◽  
Author(s):  
B.G. Pachpatte
Keyword(s):  
Asymptotic Behavior ◽  
Probability Measure ◽  
Measure Space ◽  
Discrete System ◽  
Discrete Systems ◽  
Image Position ◽  
Probability Measure Space

The object of this paper is to study a stochastic discrete system, including an operator T, of the formas a perturbation of the linear stochastic discrete systemwhere ω ∈ Ω, the supporting set of probability measure space (Ω, A, P) and n ∈ N, the set of nonnegative integers. We are concerned vith the existence, uniqueness, boundedness, and asymptotic behavior of random solutions of the above equation.

scholarly journals Notions of relative ubiquity for invariant sets of relational structures

1990 ◽  
Vol 55 (3) ◽  
pp. 948-986 ◽  
Author(s):  
Paul Bankston ◽  
Wim Ruitenburg
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
Measure Space ◽  
Invariant Sets ◽  
Natural Numbers ◽  
Compact Metric Space ◽  
Linear Orderings ◽  
Relational Structures ◽  
Probability Measure Space

AbstractGiven a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.

ERGODIC EXTENSIONS OF ENDOMORPHISMS

2015 ◽  
Vol 93 (2) ◽  
pp. 307-320
Author(s):  
EVGENIOS T. A. KAKARIADIS ◽  
JUSTIN R. PETERS
Keyword(s):  
Probability Measure ◽  
Existence And Uniqueness ◽  
Measure Space ◽  
Independent Interest ◽  
Hilbert Modules ◽  
Orthonormal Bases ◽  
Ergodic Transformations ◽  
Probability Measure Space

We examine a class of ergodic transformations on a probability measure space $(X,{\it\mu})$ and show that they extend to representations of ${\mathcal{B}}(L^{2}(X,{\it\mu}))$ that are both implemented by a Cuntz family and ergodic. This class contains several known examples, which are unified in our work. During the analysis of the existence and uniqueness of this Cuntz family, we find several results of independent interest. Most notably, we prove a decomposition of $X$ for $N$-to-one local homeomorphisms that is connected to the orthonormal bases of certain Hilbert modules.

scholarly journals Weakly mixing PET

1987 ◽  
Vol 7 (3) ◽  
pp. 337-349 ◽  
Author(s):  
V. Bergelson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Weakly Mixing ◽  
Image Position ◽  
Probability Measure Space

Suppose that (X, ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of (X, ℬ, μ). T is called weakly mixing if for any two sets A1A2 ∈ ℬ one has:

Property kα,n on Spaces with Strictly Positive Measure

1982 ◽  
Vol 34 (5) ◽  
pp. 1047-1058 ◽  
Author(s):  
S. Argyros ◽  
N. Kalamidas
Keyword(s):  
Probability Measure ◽  
Compact Space ◽  
Measure Space ◽  
Positive Measure ◽  
Negative Solution ◽  
Probability Measure Space

In this paper we study intersection properties of measurable sets with positive measure in a probability measure space, or equivalently, intersection properties of open subsets on a compact space with a strictly positive measure.The first result in this direction is due to Erdös and it is a negative solution to the problem of calibers on such spaces. In particular, under C.H., Erdös proved that Stone's space of Lebesque measurable sets of [0, 1] modulo null sets, does not have ℵ1-caliber.

Sharp ergodic theorems for group actions and strong ergodicity

1999 ◽  
Vol 19 (4) ◽  
pp. 1037-1061 ◽  
Author(s):  
ALEX FURMAN ◽  
YEHUDA SHALOM
Keyword(s):  
Probability Measure ◽  
Random Walks ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Spectral Condition ◽  
Ergodic Theorems ◽  
Averaging Operator ◽  
The Mean ◽  
Probability Measure Space

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

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