$\log$-Hölder regularity of currents and equidistribution towards Green currents

Abstract

Let $f$ be an endomorphism of a projective space or an automorphism of a compact Kähler manifold. We prove that the pull-backs of currents under the iterates of $f$ converge exponentially fast to the Green currents when tested at $\log$-Hölder-continuous observables whose $\mathrm{d}{\mathrm{d}}^{\mathrm{c}}$'s have bounded mass.