## 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**

## 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