Almost sure convergence of the clustering factor in alpha-mixing processes

Abadi and Saussol (2011) have proved that the first time a dynamical system, starting from its equilibrium measure, hits a target set A has approximately an exponential law. These results hold for systems satisfying the alpha-mixing condition with rate function alpha decreasing to zero at any rate. The parameter of the exponential law is the product lambda(A)mu(A), where the latter is the measure of the set A; only bounds for lambda(A) were given. In this note we prove that, if the rate function a decreases algebraically and if the target set is a sequence of nested cylinders sets A(n)(x) around a point x, then lambda(A(n)) converges to one for almost every point x. As a byproduct, we obtain the corresponding result for return times. (AU)