Example of an ergodic measure preserving transformation on an infinite measure space

1970 ◽  
pp. 45-52 ◽  
Author(s):  
Arshag B. Hajian ◽  
Shizuo Kakutani
Keyword(s):  
Measure Space ◽  
Ergodic Measure ◽  
Infinite Measure ◽  
Measure Preserving Transformation

Strongly equivalent invariant measures

1980 ◽  
Vol 88 (1) ◽  
pp. 33-43 ◽  
Author(s):  
J. Rosenblatt
Keyword(s):  
Measure Space ◽  
Positive Measure ◽  
Invariant Measures ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Zero Measure ◽  
Infinite Measure ◽  
Two Measures ◽  
Necessary And Sufficient

AbstractTwo measures are strongly equivalent if they have the same sets of zero measure and the same sets of infinite measure. Given a group G of strongly non-singular measurable transformations of a non-atomic positive measure space (X, β, p), if G is amenable, then a necessary and sufficient condition for there to be a G-invariant positive measure on (X, β) which is strongly equivalent to p is that p(E) > 0 implies inf p(gE) > 0 and also p(E) < ∞ implies

On the Ergodic Averages and the Ergodic Hilbert Transform

1995 ◽  
Vol 47 (2) ◽  
pp. 330-343
Author(s):  
L. M. Fernández-Cabrera ◽  
F. J. Martín-Reyes ◽  
J. L. Torrea
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Dual Problem ◽  
Finite Measure ◽  
Positive Function ◽  
Ergodic Averages ◽  
Unified Approach ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
Abstract Method

AbstractLet T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

scholarly journals Entropy of systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1118-1126
Author(s):  
RADU B. MUNTEANU
Keyword(s):  
Positive Integer ◽  
Probability Space ◽  
Bernoulli Shift ◽  
Ergodic Measure ◽  
Measure Preserving Transformation ◽  
Zero Entropy ◽  
Standard Probability

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

Mixing on Sequences

1983 ◽  
Vol 35 (2) ◽  
pp. 339-352 ◽  
Author(s):  
Nathaniel A. Friedman
Keyword(s):  
Ergodic Measure ◽  
General Definition ◽  
Weak Mixing ◽  
Independent Sequence ◽  
Density Zero ◽  
Measure Preserving Transformation ◽  
Upper Density ◽  
Mixing Sequences ◽  
Definition Of ◽  
Single Sequence

Our aim is to study the mixing sequences of a weak mixing transformation. An ergodic measure preserving transformation is weak mixing if and only if for each pair of sets there exists a sequence of density one on which the transformation mixes the sets [9]. An unpublished result of S. Kakutani implies there actually exists a single sequence of density one on which the transformation is mixing for all sets (see Section 3). This result motivated the general définition of a transformation being mixing on a sequence, as well as mixing of higher order on a sequence. Given a weak mixing transformation, there exist sequences along which it is mixing of all degrees. In particular, this is the case for an eventually independent sequence [7].In Section 3 it will be shown that if T is weak mixing but not mixing, then a sequence on which T is two-mixing must have upper density zero.

Weighted Lorentz Norm Inequalities for the One-Sided Hardy-Littlewood Maximal Functions and for the Maximal Ergodic Operator

1994 ◽  
Vol 46 (5) ◽  
pp. 1057-1072 ◽  
Author(s):  
P. Ortega Salvador
Keyword(s):  
Maximal Function ◽  
Maximal Operator ◽  
Measure Space ◽  
Maximal Functions ◽  
Norm Inequalities ◽  
Littlewood Maximal Function ◽  
Measure Preserving Transformation ◽  
The One ◽  
Image Position ◽  
Lorentz Norm

AbstractIn this paper we characterize weighted Lorentz norm inequalities for the one sided Hardy-Littlewood maximal functionSimilar questions are discussed for the maximal operator associated to an invertible measure preserving transformation of a measure space.

Ergodic Properties of Lamperti Operators, II

1983 ◽  
Vol 35 (4) ◽  
pp. 577-588 ◽  
Author(s):  
Charn-Huen Kan
Keyword(s):  
Positive Operator ◽  
Measure Space ◽  
Finite Measure ◽  
Classical Case ◽  
Variant Form ◽  
Ergodic Properties ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
General Power ◽  
Power Bounded

For T in our main Theorem 5, T* is called Lamperti in [11], whose terminology and notation we shall follow in the sequel. To avoid longish expressions, we shall also say that T* here is disjunctive and, dually, T = (T*)* is codisjunctive. The present work grows out of an attempt to establish a DEE for the general power bounded positive operator on Lp, in view of the success in the contraction case [1, 11], and forms a continuation of [11]. (In passing, we note that Calderon's technique [2] mentioned in [11] was anticipated in 1938 by M. Fukamiya [7], though in a variant form and for a more classical case, namely that of a positive Lp isometry induced by an invertible, measure preserving transformation on a totally finite measure space. Calderon's case does not assume invertibility nor total finiteness.)

A Wiener—Wintner property for the helical transform

1992 ◽  
Vol 12 (2) ◽  
pp. 185-194 ◽  
Author(s):  
I. Assani
Keyword(s):  
Dynamical System ◽  
Measure Space ◽  
Measure Preserving Transformation ◽  
Null Set

AbstractLet (X,ℱ,μ,ϕ) be a dynamical system ϕ is an invertible measure-preserving transformation on the measure space (X,ℱ,μ). We show that for each p, 1<p<∞,f ∈ Lp(μ) we can find a single null set off which exists for all ε ℝ.

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