Title: Rigidity of u-Gibbs measures for certain Anosov diffeomorphisms of the 3-torus.
Speaker: Martin Leguil (Université de Picardie Jules Verne)
Speaker's URL: https://leguil.perso.math.cnrs.fr
Abstract: We consider Anosov diffeomorphisms of the 3-torus $\mathbb T^3$ which admit a partially hyperbolic splitting $\mathbb T^3 = E^s + E^c + E^u$ such that $E^c$ is a weak unstable direction. We may consider the 2-dimensional unstable foliation $W^{cu}$ tangent to $E^c + E^u$, 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 any u-Gibbs measure is SRB.