Let f be a non-invertible holomorphic endomorphism of P k having an attracting set A. We show that, under some natural assumptions, A supports a unique invariant positive closed current τ, of the right bidegree and of mass 1. Moreover, if R is a current supported in a small neighborhood of A then its push-forwards by f n converge to τ exponentially fast. We also prove that the equilibrium measure on A is hyperbolic.