  Using alternating Heegaard diagrams, we construct some 3-manifolds which
admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms
are composed of Smale-Williams solenoid attractors and repellers, an
interesting example is the truncated-cube space. In addition, we prove that if
the nonwandering set of the diffeomorphism consists of genus two Smale-Williams
solenoids, then the Heegaard genus of the closed manifold is at most two.
