Hitting statistics for $\phi$-mixing dynamical systems

Hitting rate and escape rate are two examples of recurrence laws for a dynamical system, and a general limit connects them. We show that for both Gibbs-Markov systems or any systems with the $\phi$-mixing measure, for a sequence of nested sets whose intersection is a measure zero set, this general limit equals one in the absence of short returns and less than one otherwise, which is given by an explicit formula called extremal index. One of the applications of this result is to dynamical systems on Riemannian manifolds such as hyperbolic maps and expanding maps, and it can be applied to any system with a suitable Young tower.