Denote the set of transitive diffeos, the set of diffeo’s with Global Dominated Splittings (*GDS* for short), the set of minimal diffeos.

It is proved that

: diffeo with *GDS* can’t be minimal (here).

: robustly transitive diffeo always admits some *GDS* (here).

So , although : the special property (minimality) can’t happen in the interior of the general property (transitivity).

A minor change of the proof shows that a diffeomorphism with *GDS* can’t be uniquely ergodic, either. So we have the following conservative version:

: diffeos with *GDS* can’t be uniquely ergodic.

: stably ergodic diffeos always admits some *GDS* (here).

So , although .

**Remark.** It is a little bit tricky to define . The most natural definition may lead to an emptyset. One well-accepted definition is: if there exists a neighborhood , such that every is ergodic. All volume-preserving Anosov satisfies the later definition, and this is the context of Pugh-Shub Stable Ergodicity Conjecture.

**Remark.** There is an open dense subset , such that every is nonuniformly Anosov (here)

**Remark.** Let be a symplectic manifold with , be the set of symplectic partially hyperbolic maps with .

Then consider and be the two central Lyapunov exponents. If the dominated splitting is not refined by the partially hyperbolic splitting, then it must split the central bundle, and : is nonuniformly Anosov.

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