  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.
