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Articles 1 - 5 of 5
Full-Text Articles in Physical Sciences and Mathematics
Visually Building Smale Flows In S3, Michael C. Sullivan
Visually Building Smale Flows In S3, Michael C. Sullivan
Articles and Preprints
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, Smale flows in the 3-sphere.
Invariants Of Twist-Wise Flow Equivalence, Michael C. Sullivan
Invariants Of Twist-Wise Flow Equivalence, Michael C. Sullivan
Articles and Preprints
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.
A Zeta Function For Flows With Positive Templates, Michael C. Sullivan
A Zeta Function For Flows With Positive Templates, Michael C. Sullivan
Articles and Preprints
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.
The Prime Decomposition Of Knotted Periodic Orbits In Dynamical Systems, Michael C. Sullivan
The Prime Decomposition Of Knotted Periodic Orbits In Dynamical Systems, Michael C. Sullivan
Articles and Preprints
Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.
Composite Knots In The Figure-8 Knot Complement Can Have Any Number Of Prime Factors, Michael C. Sullivan
Composite Knots In The Figure-8 Knot Complement Can Have Any Number Of Prime Factors, Michael C. Sullivan
Articles and Preprints
We study an Anosov flow Фt in S3 – {figure-8 knots}. Birman and Williams conjectured that the knot types of the periodic orbits of this flow could have at most two prime factors. Below, we give a geometric method for constructing knots in this flow with any number of prime factors.