Tag Archives: nonuniformly hyperbolic

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.