We show that the C^2 hypothesis in the Pugh Shub ergodicity theorem for partially hyperbolic diffeomorphisms can be weakened at the cost of strengthening the center bunching hypothesis. Our results apply to partially hyperbolic diffeomorphisms whose derivatives are only H&"o;lder continuous. This is the same smoothness as required in Pesin theory.
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The research in this article was supported by National Science Foundation grant DMS-0408704. Needless to say, any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Authors' addresses: Keith Burns Department of Mathematics Northwestern University Evanston, IL 60208-2730 Amie Wilkinson Department of Mathematics Northwestern University Evanston, IL 60208-2730