Constructing equilibrium states for some partially hyperbolic attractors

We shall describe a new geometric construction of equilibrium states for a class of partially hyperbolic systems (with sub-exponential contraction in the centre-unstable direction), 'extending' the construction of u-Gibbs measures due to Pesin and Sinai. Starting with the volume on a piece of centre-unstable manifold Wᶜᵘ, if we integrate over an appropriate sequence of density functions related to some continuous potential ϕ we obtain a sequence of measures on Wᶜᵘ which are absolutely continuous with respect to Lebesgue. Pushing this sequence of measures forward and averaging, the limiting measures are equilibrium states for φ.