  We present a geometric argument that explains why some systems having
vanishing largest Lyapunov exponent have underlying dynamics aspects of which
can be effectively described by the Tsallis entropy. We rely on a comparison of
the generalised additivity of the Tsallis entropy versus the ordinary
additivity of the BGS entropy. We translate this comparison in metric terms by
using an effective hyperbolic metric on the configuration/phase space for the
Tsallis entropy versus the Euclidean one in the case of the BGS entropy.
Solving the Jacobi equation for such hyperbolic metrics effectively sets the
largest Lyapunov exponent computed with respect to the corresponding Euclidean
metric to zero. This conclusion is in agreement with all currently known
results about systems that have a simple asymptotic behaviour and are described
by the Tsallis entropy.
