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

TL;DR

This paper classifies certain Heegaard diagrams on genus two surfaces and constructs infinitely many 3-manifolds with automorphisms featuring Williams solenoid attractors and repellers, revealing diverse manifold types with stable dynamical features.

Contribution

It classifies alternating and weakly alternating Heegaard diagrams on genus two surfaces and constructs 3-manifolds with automorphisms exhibiting Williams solenoid attractors and repellers.

Findings

Constructed infinitely many genus two 3-manifolds with Williams solenoid attractors.

Included manifolds are prism manifolds, Poincaré homology sphere, and surgeries on the figure-eight knot.

Demonstrated that many 3-manifolds admit stable 'translation' automorphisms.

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-wondering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincar\'e'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.