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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Continuity of conditional measures associated to measure-preserving semiflows
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by David M. McClendon PDF
Trans. Amer. Math. Soc. 361 (2009), 331-341 Request permission

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
  • 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.
  • 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.
  • 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
  • Received by editor(s): July 27, 2006
  • Received by editor(s) in revised form: January 17, 2007
  • Published electronically: April 25, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 331-341
  • MSC (2000): Primary 37A10
  • DOI: https://doi.org/10.1090/S0002-9947-08-04501-7
  • MathSciNet review: 2439409