  The properties of the Cosmic Microwave Background (CMB) radiation must be
different in flat, positively and negatively curved universes. This fact leads
to a direct way of determining the geometry of the universe. The signature of
the predicted effect of geodesic mixing, i.e., of the `chaotic' behavior of
photon beams in negatively curved spaces peculiar to Anosov systems with strong
statistical properties, has been detected while studying the COBE-DMR 4-year
data \cite{GT}. This possible observation of the negative curvature of the
universe suggests the need to search for more effective ways to analyze the CMB
data expected from forthcoming high precision experiments. Information theory
offers such a descriptor for the CMB sky maps - the Kolmogorov complexity - as
well as provides novel insight into the problem of the loss of information and
time asymmetry in a hyperbolic universe.
