A zeta function for Smale flows that counted twisting in the local stable
manifolds of closed orbits was developed in my paper [1]. It only worked when
all the twisting was in the same sense. There exist Smale flows where
infinitely many closed orbits have the same twist. Followup papers [2,3,4] got
around this but the invariant was no long a function but an element of the
group ring ZZ/2. It generalizes the Parry-Sullivan number (that’s Denise
Sullivan). It is not known if this new invariant is commutable in general. A
joint paper with Mike Boyle [5] showed the problem to be equivalent to the
unsolved problem of classifying square matrices over ZZ/2 - which is not a PID
ring. See also [6]. 1. A zeta function for flows with positive template,
Topology & Its Applications, 66 (1995) 199-213. 2. An invariant for basic
sets of Smale flows, Ergodic Theory and Dynamical Systems, Vol 17, 1997, pp.
1437-1448. 3. Invariants of twist-wise flow equivalence, Discrete and
Continuous Dynamical Systems, Vol. 4, No. 3, July 1998, 475–484. 4.
Invariants of twist-wise flow equivalence. Electronic Research Announcements,
AMS, Vol. 3 (1997), pp. 126-130. 5. Equivariant flow equivalence of shifts of
finite type by matrix equivalence over group rings. Joint with Mike Boyle.
Proceedings of the London Mathematical Society, Volume 91 Part 1 (July 2005).
6. Twistwise flow equivalence and beyond… (Appendix joint with Mike Boyle).
The Proceedings of the Max Planck Institute Workshop on Algebraic and
Topological Dynamics, 171–186, edited by S. Kolyada, Y. Manin, & T. Ward,
Contemporary Mathematics, Vol. 385, American Mathematical Society, 2005.
Comments
A zeta function for Smale flows that counted twisting in the local stable manifolds of closed orbits was developed in my paper [1]. It only worked when all the twisting was in the same sense. There exist Smale flows where infinitely many closed orbits have the same twist. Followup papers [2,3,4] got around this but the invariant was no long a function but an element of the group ring ZZ/2. It generalizes the Parry-Sullivan number (that’s Denise Sullivan). It is not known if this new invariant is commutable in general. A joint paper with Mike Boyle [5] showed the problem to be equivalent to the unsolved problem of classifying square matrices over ZZ/2 - which is not a PID ring. See also [6]. 1. A zeta function for flows with positive template, Topology & Its Applications, 66 (1995) 199-213. 2. An invariant for basic sets of Smale flows, Ergodic Theory and Dynamical Systems, Vol 17, 1997, pp. 1437-1448. 3. Invariants of twist-wise flow equivalence, Discrete and Continuous Dynamical Systems, Vol. 4, No. 3, July 1998, 475–484. 4. Invariants of twist-wise flow equivalence. Electronic Research Announcements, AMS, Vol. 3 (1997), pp. 126-130. 5. Equivariant flow equivalence of shifts of finite type by matrix equivalence over group rings. Joint with Mike Boyle. Proceedings of the London Mathematical Society, Volume 91 Part 1 (July 2005). 6. Twistwise flow equivalence and beyond… (Appendix joint with Mike Boyle). The Proceedings of the Max Planck Institute Workshop on Algebraic and Topological Dynamics, 171–186, edited by S. Kolyada, Y. Manin, & T. Ward, Contemporary Mathematics, Vol. 385, American Mathematical Society, 2005.