Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

Accessibility and ergodicity for collapsed Anosov flows

“…See [FP,§2.3] for more discussion on this result. The case where the fundamental group is solvable is related with the existence of su-tori, see [CHHU].…”

“…First notice that since Λ is transverse to Y and Λ is closed, then Λ is uniformly transverse to Y . In addition, the second condition above implies that Λ can be completed to a foliation F so that it does not have compact leaves and F is transverse to Y (this is well known, but see explicit proofs and explanations in [FP,Lemma 3.9]). Since F does not have compact leaves it is Reebless, therefore any curve in Ă M transverse to r F intersects a leaf of r F at most once.…”

“…Many recent works including our own, have tried to show that in certain manifolds or isotopy classes ergodicity holds for all volume preserving partially hyperbolic diffeomorphisms (see [HHU 2 ,HU,GS,FP,HRHU]). Thanks to the reductions of [GPS, BuW, HHU] the problem boils down to the study of accessibility, and this has become a problem on its own (see [HP, H, HS, FP] for general results about accessibility).…”

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, II : Branching foliations

“…The tools in (5) are used in [4] to prove that a large class of partially hyperbolic diffeomorphisms in Seifert manifolds are dynamically incoherent. In addition, (2) and ( 5) are used in [18] to obtain fine dynamical consequences of partial hyperbolicity in 3-manifolds.…”