© 1996, Topology Atlas

First some rough definitions:

The chain-recurrent set of a flow are those orbits which come back near themselves infinitely often. A periodic orbit is chain recurrent, but not all chain-recurrent orbits are periodic. The set of all chain-recurrent orbits for a flow in a compact invariant set.

The chain-recurrent set is hyperbolic or has a hyperbolic structure if the tangent bundle splits into three parts: An attracting direction, a repelling direction, and the direction of the flow. For a 3-flow this implies the chain-recurrent set can be partitioned into attracting sets, repelling sets and saddle sets (Smale 67). Click here to see a piece of an orbit with a hyperbolic saddle structure. The set of orbits coming into the given orbit forms its stable manifold.

For 3-flows with hyperbolic chain-recurrent sets, the indecomposable pieces of the chain-recurrent set, called basic sets, are either single closed orbits or are "chaotic" collections of infinitely many orbits. These later objects can be modeled by templates: branched 2-manifolds with semi-flows (Birman & Williams, 83). The template is constructed from a neighborhood of the basic set by collapsing out stable manifolds. To see an example click here . This is the Lorenz template. It has two bands coming from a branch line. The semi-flow starts downward from the branch line and loops around, to the right and left, coming back to the branch line. There are many closed orbits, one of which is shown. These orbits model those in some 3-flow; they have the same knotting and lining structure.

There are at least three types of questions one can ask:

1. What knots and links reside on a given template?
2. If the template is embedded differently are there invariants we can find?
3. Using basic sets, how can they be "pasted" together to form flows on 3-manifolds? In particular we want to know how can we build nonsingular flows on the 3-sphere.

I have done work on each of these questions. I shall touch on results for the first two, and then focus on my current project addressing the third question. For an example of how to build a nonsingular flow on S^3 using a saddle set modeled by a Lorenz template click here. The knotted torus is a tubular neighborhood of an attracting closed orbit. The other torus contains an unknotted repeller. Orbits enter the neighborhood of the saddle set along the "exposed" part of its boundary. They exit the saddle set's neighborhood and flow into the neighborhood of the attractor. The neighborhood of the saddle set looks like a thickened up Lorenz template.