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, \(W^{cu}\), if we
integrate over an appropriate sequence of density functions related to
some continuous potential, \(\phi\), we obtain a sequence of measures
on \(W^{cu}\) which are absolutely continuous with respect to
Lebesgue. Pushing this sequence of measures forward and averaging, the
limiting measures are equilibrium states for \(\phi\).
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