Randomness of deterministic processes in negative curvature
A momentous legacy of twentieth-century mathematics is the
realisation that deterministically evolving systems frequently exhibit,
when observed for sufficiently extended periods of time, a statistical
behaviour akin to the limiting behaviour of independent random variables.
We shall explore a geometric incarnation of this surprising phenomenon,
overviewing various kinds of statistical limit theorems for the free motion
of a particle on a negatively curved surface. In order to emphasise the
richness of possible asymptotic behaviours, as well as the variety of
sources of randomness, we will further compare the free-motion dynamics
with a closely related evolution on the same phase space, known as the
horocycle flow.