Abstract
Pugh and Shub have conjectured that essential accessibility implies ergodicity for a $C^2$, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with $1$-dimensional center bundle. We also obtain ergodicity results for $C^{1+\delta}$ partially hyperbolic systems.
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