Alternating heegaard diagrams and williams solenoid attractors in 3-manifolds

We find all Heegaard diagrams with the property \alternating" or \weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3-manifolds, each ad- mits an automorphism whose non-wandering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincaré's homology 3-sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of 3-manifolds admit a kind of \translation" with certain stability.