  We consider discrete time dynamical systems and show the link between Hitting
Time Statistics (the distribution of the first time points land in
asymptotically small sets) and Extreme Value Theory (distribution properties of
the partial maximum of stochastic processes). This relation allows to study
Hitting Time Statistics with tools from Extreme Value Theory, and vice versa.
We apply these results to non-uniformly hyperbolic systems and prove that a
multimodal map with an absolutely continuous invariant measure must satisfy the
classical extreme value laws (with no extra condition on the speed of mixing,
for example). We also give applications of our theory to higher dimensional
examples, for which we also obtain classical extreme value laws and exponential
hitting time statistics (for balls). We extend these ideas to the subsequent
returns to asymptotically small sets, linking the Poisson statistics of both
processes.
