Problem 9

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Unique ergodicity of $W^u$ for partially Anosov diffeos.

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Something is known about minimality of strong foliations for partially hyperbolic diffeomorphisms (see Bonatti-Diaz-Ures, Journal of the IMJ, 1 (2002) 513-541).

However, minimality of the strong unstable foliation even for Anosov of T^3 with 3-bundles (2-dimensional unstable) is open.

Unique ergodicity should (?) be understood with respect to 'transverse' invariant measures. They always exist because unstable leafs have polynomial growth of volume.

Once again, I think that

Once again, I think that unique ergodicity of the unstable foliation for a volume preserving partially hyperbolic diffeomorphism may be related to the the stable ergodicity of the diffeomorphism. But this is quite speculative.