Modul:   MAT760  Ergodic theory and dynamical systems seminar

Rigidity of u-Gibbs measures for certain Anosov diffeomorphisms of the 3-torus.

Vortrag von Prof. Dr. Martin Leguil

Sprecher eingeladen von: Prof. Dr. Artur Avila

Datum: 13.03.24  Zeit: 13.30 - 14.30  Raum: ETH HG G 19.1

We consider Anosov diffeomorphisms of the 3-torus $\mathbb{T}^3$ which admit a partially hyperbolic splitting $\mathbb{T}^3 = E^s \oplus E^c \oplus E^u$ whose central direction $E^c$ is uniformly expanded. We may consider the 2-dimensional unstable foliation $W^{cu}$ tangent to $E^c \oplus E^u$, but also the 1-dimensional strong unstable foliation $W^u$ tangent to $E^u$. The behavior of $W^{cu}$ is reasonably well understood; in particular, such systems have a unique invariant measure whose disintegrations along the leaves of $W^{cu}$ are absolutely continuous: the SRB measure. The behavior of $W^u$ is less understood; we can similarly consider the class of measures whose disintegrations along the leaves of $W^u$ are absolutely continuous, the so-called u-Gibbs measures. It is well-known that the SRB measure is u-Gibbs; conversely, in a joint work with S. Alvarez, D. Obata and B. Santiago, we show that in a neighborhood of conservative systems, if the strong bundles $E^s$ and $E^u$ are not jointly integrable, then there exists a unique u-Gibbs measure, which is the SRB measure.