Alternating Heegaard diagrams and Williams solenoid attractors in $3$-manifolds

Abstract

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 admits 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.