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

Abstract: In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to H\"older potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is H\"older, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier.