Problem 126

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Put orientation into the $\zeta$ function of flows. What should $\zeta (0)$ be? Does $\zeta (0)$ depend only on $H_\ast (M_0, M_{sing}) ?$.

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A zeta function for Smale

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.