In General a Measure Preserving Transformation is Mixing

Selecta ◽  
1983 ◽  
pp. 68-74
Author(s):  
Paul R. Halmos
Keyword(s):  
Measure Preserving Transformation

Continuous Ergodic Extensions and Fibre Bundles

1978 ◽  
Vol 30 (02) ◽  
pp. 373-391 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Real Line ◽  
Lebesgue Space ◽  
Transformation Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Fibre Bundles ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Measure Preserving Transformation

If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].

Vanishing transverse entropy in smooth ergodic theory

2010 ◽  
Vol 31 (4) ◽  
pp. 1229-1235 ◽  
Author(s):  
FRANÇOIS LEDRAPPIER ◽  
JIAN-SHENG XIE
Keyword(s):  
Ergodic Theory ◽  
Measure Preserving Transformation ◽  
Smooth Ergodic Theory

AbstractFor a measure-preserving transformation, the entropy being zero means that there is no increasingσ-algebra. In this note, we prove that a similar phenomenon occurs forC2diffeomorphisms when considering the increment between the partial entropies associated with different exponents.

Weak-type inequalities and convergence of the ergodic averages in variable Lebesgue spaces with weights

2009 ◽  
Vol 139 (4) ◽  
pp. 673-683 ◽  
Author(s):  
M. Isabel Aguilar Cañestro ◽  
Pedro Ortega Salvador
Keyword(s):  
Lebesgue Space ◽  
Weak Type ◽  
Suitable Condition ◽  
Variable Exponents ◽  
Ergodic Averages ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Almost Everywhere ◽  
Variable Lebesgue Space ◽  
Measure Preserving Transformation

We characterize the weighted weak-type inequalities with variable exponents for the maximal operator associated with an ergodic, invertible, measure-preserving transformation and prove the almost everywhere convergence of the ergodic averages for all functions in a variable Lebesgue space with a weight verifying a suitable condition.

scholarly journals A mixing dynamical system on the cantor set

1995 ◽  
Vol 18 (3) ◽  
pp. 607-612
Author(s):  
Jeong H. Kim
Keyword(s):  
Dynamical System ◽  
Cantor Set ◽  
Strong Mixing ◽  
Mixing Properties ◽  
Measure Preserving Transformation ◽  
Shift Map ◽  
The One

In this paper we give mixing properties (ergodic, weak-mixng and strong-mixing) to a dynamical system on the Cantor set by showing that the one-sided(12,12)-shift map is isomorphic to a measure preserving transformation defined on the Cantor set

scholarly journals A return time invariant for finitary isomorphisms

1984 ◽  
Vol 4 (2) ◽  
pp. 225-231 ◽  
Author(s):  
U. R. Fiebig
Keyword(s):  
Finite Measure ◽  
Measurable Subset ◽  
Return Time ◽  
Time Function ◽  
Markov Shifts ◽  
Measure Preserving Transformation ◽  
Recurrence Theorem ◽  
First Return Time ◽  
Time Invariant ◽  
Finitary Isomorphisms

AbstractPoincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

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.

The Pointwise Ergodic Theorem for Transformations whose Orbits contain or are contained in the Orbits of a Measure-Preserving Transformation

1982 ◽  
Vol 34 (6) ◽  
pp. 1303-1318 ◽  
Author(s):  
John C. Kieffer ◽  
Maurice Rahe
Keyword(s):  
Information Theory ◽  
Ergodic Theory ◽  
Probability Space ◽  
Sufficient Conditions ◽  
Measurable Transformation ◽  
Input And Output ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Measure Preserving Transformations

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

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