  It is well known that the so called Bianchi IX spacetimes with SO(3)-symmetry
in a neighbourhood of the Big Bang exhibit a chaotic behaviour of typical
trajectories in the backward movement of time. This behaviour (Mixmaster Model
of the Universe) can be encoded by the shift of two-sided continued fractions.
Exactly the same shift encodes the sequences of intersections of hyperbolic
geodesics with purely imaginary axis in the upper complex half-plane, that is
geodesic flow on an appropriate modular surface. A physical interpretation of
this coincidence was suggested in arXiv:1402.2158: namely, that Mixmaster chaos
is an approximate description of the passage from a hot quantum Universe at the
Big Bang moment to the cooling classical Universe. Here we discuss and
elaborate this suggestion, looking at the Mixmaster Model from the perspective
of the second class of Bianchi IX spacetimes: those with SU(2)-symmetry
(self-dual Einstein metrics). We also extend it to the more general context
related to Painleve' VI equations.
