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On the ergodicity of partially hyperbolic systems. (English) Zbl 1196.37057
A partially hyperbolic diffeomorphism of a compact manifold \(M\) is called accessible if any point of \(M\) can be reached from any other point along a path that is a concatenation of finitely many subpaths each of which lies entirely in a single leaf of the strong stable (or unstable) foliation. A diffeomorphism \(f\) is called essentially accessible if every measurable set that is a union of entire accessibility classes has either full or zero measure. It was conjectured by C. Pugh and M. Shub [J. Complexity 13, No. 1, 125–179 (1997; Zbl 0883.58025)] that if \(f\) is a \(C^2\), partially hyperbolic, volume preserving diffeomorphism, then essential accessibility of \(f\) implies ergodicity. The authors prove this conjecture under a so-called center bunching condition (which is satisfied, for example, if the center bundle is one-dimensional).

37D30 Partially hyperbolic systems and dominated splittings
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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