## Atomic disintegrations for partially hyperbolic diffeomorphisms

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## Abstract:

Shub and Wilkinson and Ruelle and Wilkinson studied a class of volume preserving diffeomorphisms on the three dimensional torus that are stably ergodic. The diffeomorphisms are partially hyperbolic and admit an invariant central foliation of circles. The foliation is not absolutely continuous; in fact, Ruelle and Wilkinson established that the disintegration of volume along central leaves is atomic. We show that in such a class of volume preserving diffeomorphisms the disintegration of volume along central leaves is a single delta measure. We also formulate a general result for conservative three dimensional skew product like diffeomorphisms on circle bundles, providing conditions for delta measures as disintegrations of the smooth invariant measure.## References

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## Additional Information

**Ale Jan Homburg**- Affiliation: KdV Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands – and – Department of Mathematics, VU University Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, Netherlands
- MR Author ID: 356749
- Email: a.j.homburg@uva.nl
- Received by editor(s): September 29, 2015
- Received by editor(s) in revised form: March 7, 2016, and August 8, 2016
- Published electronically: January 6, 2017
- Communicated by: Nimish Shah
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 2981-2996 - MSC (2010): Primary 37C05, 37D30
- DOI: https://doi.org/10.1090/proc/13509
- MathSciNet review: 3637946