For a partially hyperbolic diffeomorphism f, a Gibbs u-state is a measure whose decomposition along unstable leaves are absolutely continuous with respect to Lebesgue measure. In this talk we consider ʋ Gibbs states: measures whose conditional measures are absolutely continuous with a given reference measure ʋ (not necessarily Lebesgue). We will discuss their existence, finiteness, structure of the supports, and how to obtain decay of correlations even when the system does not have enough regularity.

This is a joint work with Raul Ures, Marcelo Viana and Jiagang Yang.