Visually building Smale flows in S3

A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere

Knots on a positive template have a bounded number of prime factors

Templates are branched 2-manifolds with semi-flows used to model `chaotic' hyperbolic invariant sets of flows on 3-manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be bounded. We prove a special case when the template is positive; the general case is now known to be false.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-24.abs.htm

Invariants of Twist-wise Flow Equivalence

Flow equivalence of irreducible nontrivial square nonnegative integer matrices is completely determined by two computable invariants, the Parry-Sullivan number and the Bowen-Franks group. Twist-wise flow equivalence is a natural generalization that takes account of twisting in the local stable manifold of the orbits of a flow. Two new invariants in this category are established

Educating Dilbert

A commentary on calculus reform efforts

Factoring Families of Positive Knots on Lorenz-like Templates

We show that for m and n positive, composite closed orbits realized on the Lorenz-like template L(m, n) have two prime factors, each a torus knot; and that composite closed orbits on L(−1,−1) have either two for three prime factors, two of which are torus knots

Factoring Positive Braids via Branched Manifolds

We show that a positive braid is composite if and only if the factorization is visually obvious by placing the braid k in a specially constructed smooth branched 2- manifold B(k) and studying how a would-be cutting sphere meets B(k). This gives an elementary proof of a theorem due to Peter Cromwell

Symbolic Dynamics and its Applications

Book review of Symbolic Dynamics and its Applications, edited by Susan Williams, AMS

Twistwise Flow Equivalence and Beyond...

An expository account of recent progress on twistwise flow equivalence. There is a new result in the appendix. (Appendix joint with Mike Boyle.

A Zeta Function for Flows with Positive Templates

A zeta function for a map f : M → M is a device for counting periodic orbits. For a topological flow however, there is not a clear meaning to the period of a closed orbit. We circumvent this for flows which have positive templates by counting the “twists” in the stable manifolds of the periodic orbits