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In this paper we prove that for topologically mixing Anosov flows theirequilibrium states corresponding to H\"older potentials satisfy a strongrigidity property: they are determined only by their disintegrations on(strong) stable or unstable leaves.  As a consequence we deduce: the corresponding horocyclic foliations of suchsystems are uniquely quasi-ergodic, provided that the corresponding Jacobian isH\"older, without any restriction on the dimension of the invariantdistributions. This generalizes a classical result due to Babillott andLedrappier for the geodesic flow of hyperbolic manifolds.  We rely on symbolic dynamics and on recent methods developed by the authors.

Rigidity of equilibrium states and unique quasi-ergodicity for horocyclic foliations