Let
be a closed
–manifold
which admits an Anosov flow. We develop a technique for constructing
partially hyperbolic representatives in many mapping classes of
.
We apply this technique both in the setting of geodesic flows on closed
hyperbolic surfaces and for Anosov flows which admit transverse tori. We
emphasize the similarity of both constructions through the concept of
–transversality,
a tool which allows us to compose different mapping classes while retaining partial
hyperbolicity.
In the case of the geodesic flow of a closed hyperbolic surface
we build stably ergodic, partially hyperbolic diffeomorphisms whose
mapping classes form a subgroup of the mapping class group
which is
isomorphic to
.
At the same time we show that the totality of mapping classes which can be
realized by partially hyperbolic diffeomorphisms does not form a subgroup of
.
Finally, some of the examples on
are absolutely partially hyperbolic, stably ergodic and robustly nondynamically
coherent, disproving a conjecture of Rodriguez Hertz, Rodriguez Hertz and Ures
(Ann. Inst. H Poincaré Anal. Non Linéaire 33 (2016) 1023–1032).