$\log$-H\"older regularity of currents and equidistribution towards Green currents

Abstract

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