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Transactions of the American Mathematical Society

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Continuity of conditional measures associated to measure-preserving semiflows


Author: David M. McClendon
Journal: Trans. Amer. Math. Soc. 361 (2009), 331-341
MSC (2000): Primary 37A10
Published electronically: April 25, 2008
MathSciNet review: 2439409
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Abstract: Let $ X$ be a standard probability space and $ T_t$ a measure-preserving semiflow on $ X$. We show that there exists a set $ X_0$ of full measure in $ X$ such that for any $ x \in X_0$ and $ t \geq 0$ there are measures $ \mu_{x,t}^+$ and $ \mu_{x,t}^-$ which for all but a countable number of $ t$ give a distribution on the set of points $ y$ such that $ T_t(y) = T_t(x)$. These measures arise by taking weak$ ^*-$limits of suitable conditional expectations. Say that a point $ x$ has a measurable orbit discontinuity at time $ t_0$ if either $ \mu_{x,t}^+$ or $ \mu_{x,t}^-$ is weak$ ^*-$discontinuous in $ t$ at $ t_0$. We show that there exists an invariant set of full measure in $ X$ such that any point in this set has at most countably many measurable orbit discontinuities. Furthermore we show that if $ x$ has a measurable orbit discontinuity at time 0, then $ x$ has an orbit discontinuity at time 0 in the sense of Orbit discontinuities and topological models for Bordel semiflows, D. McClendon.


References [Enhancements On Off] (What's this?)

  • 1. D. McClendon, Orbit discontinuities and topological models for Borel semiflows, submitted to Erg. Th. & Dyn. Sys., available at www.math.northwestern.edu/$ ^\sim$dmm/semiflowpaper1.pdf, 2007.
  • 2. D. McClendon, Universally modeling Borel semiflows by a shift action on a space of left-continuous functions, preprint, available at www.math.northwestern.edu/$ ^\sim$dmm/pathmodel. pdf, 2007.
  • 3. Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631

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Additional Information

David M. McClendon
Affiliation: Department of Mathematics, University of Maryland at College Park, College Park, Maryland 20742-4015
Address at time of publication: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370
Email: dmm@math.northwestern.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04501-7
Keywords: Measure-preserving semiflows
Received by editor(s): July 27, 2006
Received by editor(s) in revised form: January 17, 2007
Published electronically: April 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.