The Linking Homomorphism of Onedimensional Minimal Sets
Alex Clark and Michael C. Sullivan
Abstract:
We introduce a way of characterizing the linking of onedimensional minimal
sets in threedimensional flows and carry out the characterization for some
minimal sets within flows modeled by templates.

The Paper
Minimal Sets
If f is a continuous flow or map on a space X,
then a closed subset M of X is a minimal set
of f if M is invariant under f but contains
no proper, nonempty, closed set which is also invariant. Periodic orbits
are minmal sets, but more complicated minimal sets exist.
References:
 Qualitative Theory of Differential Equations, Nemytskii & Stepanov,
Princeton University Press, 1960.
 Lectures on Topological Dynamics, Robert Ellis, Benjamin, Inc., 1969.
Sturmian Minimal Sets
For each irrational number a
in the open interval (0,1), a Strumian minimal set can be generated
by the following algorythm. Let t be a real number.
Let b_{n} = t+
na mod 1. Then define
x_{n} to be 0 when b_{n}
is less than or equal to a or 1 otherwise.
Then (x_{n})_{n>0} is a (onesided)
Sturmian minimal set. Sturmian minimal sets have "minimal block growth."
Thus, they are the simplest minimal sets after periodic orbits.
References:
 Gottschalk, W. H., Hedlund, G., Topological
dynamics, A. M. S. Colloquium Publications, Vol. 36., 1955.
 Coven, E. M.; Hedlund, G. A., Sequences with
minimal block growth, Math. Systems Theory 7 (1973), 138153.
 Paul, M. E., Minimal symbolic flows having minimal
block growth, Math. Systems Theory 8 (1974/75), no. 4, 309315.
Below is a maple procedure for generating Sturmian minimal sets:
>
sturmian:=proc(alpha, t, numterms)
local i, word;
word:=[];
for i from 1 to numterms do
if (evalf(frac(t+i*alpha)) <= evalf(alpha)) then word:=[op(word),0] else word:=[op(word),1] fi;
od;
word;
end:
>
sturmian(sqrt(1/2),0, 100);
Lorenz Template
The Lorenz template is a branched 2manifold with a semiflow that gives a
geometic model of the famous Lorenz attractor. Orbits in the Lorenz template
are associated to the full 2shift, and so contain Sturmian minimal sets.
Refernces:
 Birman, J. S.; Williams, R. F., Knotted periodic
orbits in dynamical systems. I. Lorenz's equations. Topology 22 (1983), no.
1, 4782.
 Birman, J. S.; Williams, R. F., Knotted periodic
orbits in dynamical system. II. Knot holders for fibered knots,
Contemp. Math., 20, Amer. Math. Soc., (1983), 160.
 Ghrist, R.; Holmes, P. J.; Sullivan, M. C., Knots
and links in threedimensional flows. Lecture Notes in Mathematics, 1654
(1997), SpringerVerlag, Berlin.
We have used a Maple procedure to plot the Sturmian minimal set above
in the Lorenz template.
Linking
In the above figure the minimal set appears to live in a small neighborhood
of a wedge of two circles, a figure8. The homology calculations in our paper confirm this.
This allows one to talk about how two Sturmian minimal sets are linked. But
instead of a linking number one gets a 2x2 linking matrix. Because one has
some freedom in choosing bases for the neighborhoods the matrix itself is not
an invariant, but its Smith normal form is. (See the ismith command in Maple.)
Below is a plot of two Sturmian minimal sets. The the green one
has a =1/sqrt(2), the blue one has a=1/sqrt(11). Both have t=0.
Careful inspection gives a preliminary linking matrix , whose Smith normal form is .
Other plots not in paper:
Maple programs used to make plots
Alex here are the other programs I have: