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

Chao Wang, Yimu Zhang



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.


Heegaard diagram; solenoid attractor; Prism manifold; Poincaré's homology $3$-sphere; figure eight knot

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