  We use numerical and analytical tools to demonstrate arguments in favor of
the existence of a family of smooth, symplectic diffeomorphisms of the
two-dimensional torus that have both a positive measure set with positive
Lyapunov exponent and a positive measure set with zero Lyapunov exponent. The
family we study is the unfolding of an almost-hyperbolic diffeomorphism on the
boundary of the set of Anosov diffeomorphisms, proposed by Lewowicz.
