Given a general Anosov abelian action on a closed manifold, we study properties of certain invariant measures that have recently been introduced in cite{BGHW20} using the theory of Ruelle-Taylor resonances. We show that these measures share many properties of Sinai-Ruelle-Bowen measures for general Anosov flows such as smooth desintegrations along the unstable foliation, positive Lebesgue measure basins of attraction and a Bowen formula in terms of periodic orbits.
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