Martin Leguill: On the uniqueness of u-Gibbs measures for Anosov diffeomorphisms of the 3-torus

Speaker
Martin Leguill, École Polytechnique

Abstract
We consider an Anosov diffeomorphism \( f \) of the 3-torus \( \mathbb{T}^3 \) with a partially hyperbolic splitting \( E^s \oplus E^c \oplus E^u \), where the central bundle \( E^c \) is uniformly expanding. U-Gibbs measures are \( f \)-invariant measures whose conditionals along the strong unstable foliation \( W^u \) are absolutely continuous; they capture all possible statistical behaviors for a set of initial conditions of full volume. In earlier work with S. Alvarez, D. Obata, and B. Santiago, we showed that if \( f \) is weakly dissipative and \( E^s, E^u \) are not jointly integrable, then there exists a unique u-Gibbs probability measure: the SRB measure. More recently, in collaboration with S. Crovisier and the aforementioned co-authors, we consider the complementary case where \( E^s, E^u \) are jointly integrable; we show that even in this case, there exists a unique u-Gibbs measure (with non-zero transverse entropy). The proof relies in particular on the construction of a horocyclic flow compatible with the u-Gibbs measures, whose ergodic properties we study.