scholarly journals Ergodicity Space for Measure-Preserving Transformations

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
M. Rahimi ◽  
A. Assari
Keyword(s):  
Dynamical Systems ◽  
Measure Preserving Transformation ◽  
Measure Preserving Transformations

We introduce the concept of ergodicity space of a measure-preserving transformation and will present some of its properties as an algebraic weight for measuring the size of the ergodicity of a measure-preserving transformation. We will also prove the invariance of the ergodicity space under conjugacy of dynamical systems.

scholarly journals Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 309
Author(s):  
Xianfeng Ma ◽  
Zhongyue Wang ◽  
Hailin Tan
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Random Dynamical Systems ◽  
Topological Pressure ◽  
Image Measure ◽  
Measure Preserving Transformations

A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.

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.

$C^\ast$-Algebras of singular integral operators and Toeplitz operators associated with measure-preserving transformations

1998 ◽  
Vol 18 (2) ◽  
pp. 487-502
Author(s):  
JINGBO XIA
Keyword(s):  
Dynamical Systems ◽  
Singular Integral ◽  
Toeplitz Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Measure Preserving Transformations

We show in the setting of measure-theoretical dynamical systems that certain singular integral operators and Toeplitz operators with the same coefficients but represented on different $L^2$-spaces are $C^\ast $-algebraically equivalent.

Spectral Properties for Invertible Measure Preserving Transformations

1973 ◽  
Vol 25 (4) ◽  
pp. 806-811 ◽  
Author(s):  
Jean-Marc Belley
Keyword(s):  
Spectral Properties ◽  
Unitary Operator ◽  
Unit Interval ◽  
Complex Conjugation ◽  
Integrable Functions ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Square Integrable Functions ◽  
Measure Preserving Transformations

An invertible measure preserving transformation T on the unit interval I generates a unitary operator U on the space L2(I) of Lebesque square integrable functions given by (Uf)(x) = f(Tx) for all f in L2(I) and x in I. By definitionfor all f , g in L2(I), the bar denoting complex conjugation.

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