Tag Archives: stably ergodic

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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dominated nonuniformly hyperbolic

Bochi, Fayad, Pujals, A remark on conservative diffeomorphisms.

Let r>1, \mathrm{Diff}^r_\mu(M) endow with C^1 topology (which is not a complete metric space). A map f\in\mathrm{Diff}^r_\mu(M) is said to be dominated nonuniformly hyperbolic if there exist a dominated splitting TM=E^+\oplus E^-, where E^+ (resp. E^-) coincides a.e. with the sum of the Oseledets spaces corresponding to positive (resp. negative) Lyapunov exponents. Let \mathrm{SE}^r be the stably ergodic ones in \mathrm{Diff}^r(M). Clearly the DNH is an open property in \mathrm{SE}^r(M) and may not be open in \mathrm{Diff}^r(M)..

Theorem. There is an open and dense set \mathcal{R}^r\subset\mathrm{SE}^r such that each f\in\mathcal{R}^r is dominated nonuniformly hyperbolic. In particular the stably DNH are C^1 dense in \mathrm{SE}^r.

In fact \mathcal{R}^r is the set of f \in \mathrm{SE}^r such that f has a dominated splitting TM=E^+\oplus E^- with \lambda_p(f ) > 0 > \lambda_{p+1}(f ), where p = \dim E^+.

It is not true that every stably ergodic diffeomorphism can be approximated by a partially hyperbolic system.

1. A stably ergodic (or stably transitive) diffeomorphism f must have a dominated splitting. This is true because if it did not, they perturbed f and create a periodic point whose derivative is the identity. Then, using the Pasting lemma (for which r>1 regularity is an essential hypothesis), one breaks transitivity.

2. They further perturbed f such that the sum of the Lyapunov exponents `inside’ each of the bundles of the (finest) dominated splitting is non-zero.

3. They finally perturbed f such that the Lyapunov exponents in each of the bundles become almost equal. (If one attempted to make the exponents exactly equal, there is no guarantee that the perturbation is C^r.) Since the sum of the exponents in each bundle varies continuously, they concluded there are no zero exponents.

The continuation of the finest dominated splitting is not necessarily the finest dominated splitting of the perturbed diffeomorphism. A dominated splitting is stably finest if it has a continuation which is the finest dominated splitting of every sufficiently close diffeomorphism. It is easy to see that diffeomorphisms with stably finest dominated splittings are (open and) dense among diffeomorphisms with a dominated splitting.