Some remarks about dominated splitting property

Denote \mathcal{T} the set of transitive diffeos, \mathcal{DS} the set of diffeo’s with Global Dominated Splittings (GDS for short), \mathcal{M} the set of minimal diffeos.

It is proved that

\mathcal{DS}\bigcap \mathcal{M}=\emptyset: diffeo with GDS can’t be minimal (here).

\mathcal{T}^o\subset \mathcal{DS}: robustly transitive diffeo always admits some GDS (here).

So \mathcal{T}^o\bigcap \mathcal{M}=\emptyset, although \mathcal{T}\supset \mathcal{M}: 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:

\mathcal{DS}\bigcap \mathcal{UE}=\emptyset: diffeos with GDS can’t be uniquely ergodic.

\mathcal{E}^o\subset \mathcal{DS}: stably ergodic diffeos always admits some GDS (here).

So \mathcal{E}^o\bigcap \mathcal{UE}=\emptyset, although \mathcal{E}\supset \mathcal{UE}.

Remark. It is a little bit tricky to define \mathcal{E}^o. The most natural definition may lead to an emptyset. One well-accepted definition is: f\in\mathcal{E}^o if there exists a C^1 neighborhood f\in\mathcal{U}\subset\mathrm{Diff}^1_m(M), such that every g\in \mathcal{U}\cap \mathrm{Diff}^2_m(M) 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 \mathcal{R}\subset \mathcal{E}^o, such that every f\in \mathcal{R} is nonuniformly Anosov (here)

Remark. Let (M,\omega) be a symplectic manifold with \dim M\ge 4, \mathrm{PH}^{2}_{\omega}(M,2) be the set of C^2 symplectic partially hyperbolic maps with \dim (E^c)=2.
Then consider f\in\mathcal{E}^o\cap \mathrm{PH}^{2}_{\omega}(M,2) and \lambda^c_1(f,\omega)\ge \lambda^c_2(f,\omega) 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 \lambda^c_1(f,\omega)> \lambda^c_2(f,\omega): f is nonuniformly Anosov.

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