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## A dichotomy for volume-preserving diffeomorphisms in dimension 3

On December 3rd, 2009, M.A. Rodriguez-Hertz presented at the seminar of “Topologie et dynamique” at Orsay a new result extending to three dimensional manifolds the well-known theorem announced by Mañé and proved by Bochi in 2002:

Theorem (M.A. Rodriguez-Hertz). Consider the space of all C^1 diffeomorphism preserving the volume on a three-dimensional compact manifold. Then generically, one of the following occurs:

• all Lyapunov exponents vanish Lebesgue-almost everywhere;
• there is a dominated splitting and Lebesgue-almost everywhere it separates positive and negative exponents (and there is no zero Lyapunov exponents) and it is ergodic with respect to volume.

## Ergodicity of partially hyperbolic symplectomorphisms

Amie WILKINSON presented new results towards the Pugh-Shub Stable ergodicity conjecture. In particular, with A. AVILA and J. BOCHI, she proved that ergodicity is generic in C^1 partially hyperbolic symplectomorphisms. She noted that, by a result of SAGHIN and Z. XIA, a stably ergodic symplectomorphism is automatically partially hyperbolic (which fails for conservative diffeomorphisms by an example of A. TAHZIBI).

The ingredients of the proof are:

• Bochi’s alternative: a symplectomorphism which is not Anosov has only zero exponents Lebesgue almost everywhere.
• A new criterion for ergodicity and saturation for C^2 partially hyperbolic diffeomorphisms  involving a new, non-uniform variant of the center bunching property (used to prove that a set is simultaneously saturated wrt to the stable and unstable foliations)
• A perturbation technique introduced by M.-C. ARNAUD in her proof of Mañé’s ergodic closing lemma
• The ergodic  diffeomorphisms of the disk arbitrarily close to the identity, built by ANOSOV and KATOK
• A Baire argument

## Stable ergodicity and partial hyperbolicity

Ya. Pesin gave a talk (Chevaleret, June 19, 2009) summarizing the work on Push-Shub Stable Ergodicity Conjecture and presenting some new results of his in the non-conservative case.

This conjecture states that there is a $C^1$-dense and open set of ergodic diffeomorphisms among those which are partially hyperbolic and volume preserving. Note the tension between perturbative techniques which are mainly known in the $C^1$ setting and smooth ergodic theory in the $C^2$ setting.

One tries to establish accessibility, i.e., that it is possible to link any two points following alternatively stable and unstable manifolds. This is now known to hold:

• $C^1$-densely according to Dolgopyat and Wilkinson;
• if the central dimension is 1, $C^r$-densely according to Didier ($r=2$) and Rodriguez-Hertz, Roderiguez-Hertz and Ures.

Ergodicity then follows under $C^2$ smoothness and center-bunching (i.e., near conformality of the central bundle) according to Burns and Wilkinson and Rodriguez-Hertz, Roderiguez-Hertz and Ures.

The talk then turned to the non-conservative case, first recalling the result of Burns, Dolgopyat and Pesin establishing stable ergodicity provided: (i) all the exponents are strictly negative on a set of positive volume; (ii) $C^2$ smoothness; (iii) essential accessibility.

More precisely, $f$ is a partially hyperbolic, $C^2$ diffeomorphism of a compact manifold with a topological attractor $\Lambda$. The role of the volume (no longer preserved) is played by u-measures.

They have been built “from the outside” by pushing the volume by Pesin-Sinai and Bonatti-Diaz-Viana. They have been characterized “from the inside”, by pushing the leaf volume by Pesin-Sinai. According to Bonatti-Diaz-Viana, any physical measure is a u-measure. According to Dolgopyat, a unique u-measure is a physical measure.

It is not difficult to see that uniqueness of the u-measure may fail even for topologically transitive attractor.

Pesin stated three theorems:

Theorem A.

1. There exists $\mu$, a u-measure as soon as the Lyapunov exponents along the central bundle are all strictly negative and all unstable leaves are dense in the attractor.
2. if $\mu$ is unique as a u-measure, it is unique as an S.R.B. measure.
3. $\mu$ is Bernoulli.

This is related to results of Alvès-Bonatti-Viana. It applies to some examples with central dimension 1 (otherwise the sum of the central exponents is negative and this is a problem).

Theorem B.

In the setting of Theorem A, any $C^2$-diffeomorphism $g$ which is $C^1$-close to $f$:

1. automatically satisfies the negativity of its central exponents;
2. has a unique u-measure $\mu$ (which is SRB);
3. this $\mu$ has full basin and is Bernoulli.

It is interesting to note that this theorem fails if one instead assumes that the central exponents are strictly positive. The authors do not claim that all unstable leaves are dense.

Theorem C.

Let $f$ be a $C^2$-diffeomorphism with a partially hyperbolic attractor $\Lambda_f$. Assume (i) there is a unique u-measure with strictly positive central exponents $\mu$-almost everywhere; (ii) all unstable leaves are dense in $\Lambda_f$. Then $f$ is stably ergodic.

This uses the result by Dolgopyat that the limit of a sequence of u-measures for a sequence of partially hyperbolic diffeomorphisms is itself a u-measure for the limit diffeomorphism (when they both exist). It also uses Pliss Lemma to get mesoscopic local central disk.

It is noted that they are few results in the presence of critical points (see Viana’s multidimensional attractors, however).

It is also observed that integrability is known to be breakable by a $C^1$-perturbation (recall the result of Dolgopyat-Wilkinson). This is unknown in the $C^2$-topology, except for central dimension 1 where a result of JP Marco exists.