In this paper we provide a connection between the geometrical properties of a
chaotic dynamical system and the distribution of extreme values. We show that
the extremes of so-called physical observables are distributed according to the
classical generalised Pareto distribution and derive explicit expressions for
the scaling and the shape parameter. In particular, we derive that the shape
parameter does not depend on the chosen observables, but only on the partial
dimensions of the invariant measure on the stable, unstable, and neutral
manifolds. The shape parameter is negative and is close to zero when
high-dimensional systems are considered. This result agrees with what was
derived recently using the generalized extreme value approach. Combining the
results obtained using such physical observables and the properties of the
extremes of distance observables, it is possible to derive estimates of the
partial dimensions of the attractor along the stable and the unstable
directions of the flow. Moreover, by writing the shape parameter in terms of
moments of the extremes of the considered observable and by using linear
response theory, we relate the sensitivity to perturbations of the shape
parameter to the sensitivity of the moments, of the partial dimensions, and of
the Kaplan-Yorke dimension of the attractor. Preliminary numerical
investigations provide encouraging results on the applicability of the theory
presented here. The results presented here do not apply for all combinations of
Axiom A systems and observables, but the breakdown seems to be related to very
special geometrical configurations.Comment: 16 pages, 3 Figure