Extreme Value Analysis in dynamical systems: two case studies

We give here a brief summary of classical Extreme Value Theory for random variables, followed by that for deterministic dynamical systems, which is a rapidly developing area of research. Here we would like to contribute to that by conducting a numerical analysis designed to show particular features of extreme value statistics in dynamical systems, and also to explore the validity of the theory. We find that formulae that link the extreme value statistics with geometrical properties of the attractor hold typically for high-dimensional systems – whether a so-called geometric distance observable or a physical observable is concerned. In very low-dimensional settings, however, the fractality of the attractor prevents the system from having an extreme value law, which might well render the evaluation of extreme value statistics meaningless and so ill-suited for application.