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.
shub (2017-07-10 09:59:28 -0700 -0700):
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.
Comments
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 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.