Measure Preserving Systems
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Entropy of infinite systems and transformations
The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...
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Dirac measure is an important measure in many related branches to mathematics. The current paper characterizes measure-preserving transformations between two Dirac measure spaces or a Dirac measure space and a probability measure space. Also, it studies isomorphic Dirac measure spaces, equivalence Dirac measure algebras, and conjugate of Dirac measure spaces. The equivalence classes of a Dirac ...
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These notes are from a graduate course given in Spring 2007 at the University of North Carolina at Chapel Hill. A major portion will be published as part of the Springer Online Encyclopedia on Complexity. They provide an introduction to the subject of measure-preserving dynamical systems, discussing the dynamical viewpoint; how and from where measure-preserving systems arise; the construction o...
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We prove that an extension of an invertible, multiply-recurrent infinite measure preserving transformation is also multiply-recurrent.
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