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Twist-wise flow equivalence
by
Michael C. Sullivan
Southern Illinois University at Carbondale
A self homeomorphism the Cantor set, h:C --> C, can be suspended to produce a flow on a one-dimensional space. (Such objects arise as basic sets of Smale flows on a manifold.) Two such flows are topologically equivalent if there is an orbit preserving homeomorphism bewtween them. A square nonnegative integral matrix can be associated to a Markov partition for h. Two such matrices are said to be flow equivalent if they give rise to topologically equivalent suspension flows. John Franks has devised a complete set of computable invariants to dedect flow equivalence of irreducable square nonnegative integral matrices.
We ask what happens when the matrix encodes additional information about h. In particular, orietation data is considered: in Smale flows there are stable and unstable manifolds for each orbit in a basic set. These can twist around. This twisting data is encoded in matrices with entries of the form a+bt, where a and b are nonnegative integers and all calculations are done modulo t2=1. Then one can define twist-wise flow equivalence of such matrices. Several invariants will be defined, but they are not complete.
Date received: April 27, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-08.