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Lefschetz formulas for flows. (English) Zbl 0619.58034
Differential equations, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, Pt. III, Contemp. Math. 58, No. 3, 19-69 (1987).
[For the entire collection see Zbl 0607.00012.]
According to the known Wilson plug construction every nonsingular flow $$\phi$$ may be deformed through nonsingular flows to a flow $$\psi$$ with no closed orbits. So there are no correlations between the existence of closed orbits of the nonsingular flow $$\phi$$ and the topology of the underlying manifold M. The author shows, nonetheless, that for many types of $$\phi$$ there is a quantitative relation between closed orbits and the underlying topology of M. This relation the author calls Lefschetz formula for $$\phi$$. Namely: let E be a flat vector bundle over M, $$\phi_ E(\gamma)$$ holonomy of E around the loop $$\gamma$$. Then zeta function may be introduced as: $Z_{Q,E}(S)=\prod_{\gamma}(\det I- \Delta (\gamma)\rho_ E(\gamma)e^{-Sl(\gamma)})^{(-1)^{u(\gamma)}}$ where $$\gamma$$ runs over all prime closed orbits of $$\phi$$, u($$\gamma)$$ is the dimension of the unstable bundle over $$\gamma$$, $$\Delta (\gamma)=\pm 1$$ as this bundle is orientable or not. Let us note, also, that $$\psi_{\phi,E}=| Z_{\phi,E}(0)|$$ and $$\tau$$ (E) is l-torsion (Reidemeister torsion). Then according to the author the pair ($$\phi$$,E) is called Lefschetz if $$\psi_{\phi,E}=\tau (E)$$. One of the main statements of the article is the following: ”If the chain recurrent set R($$\phi)$$ is circular (i.e. admits cross-sections) then for certain bundles E, ($$\phi$$,E) is Lefschetz” (§ 3). Some other results may be mentioned: if X has constant positive curvature, then any acyclic bundle over X is Lefschetz (cor. 6.2). For oriented X of constant negative curvature, any acyclic unitary E over SX is Lefschetz (Theorem 6.5, where SX is a unit sphere bundle of X). At the end of the paper the author conjectures that: if X is a compact locally homogeneous Riemannian manifold and E is an acyclic bundle over SX then E is Lefschetz.
Reviewer: V.B.Marenich

##### MSC:
 37C10 Dynamics induced by flows and semiflows 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 37G99 Local and nonlocal bifurcation theory for dynamical systems
##### Keywords:
Wilson plug; nonsingular flow; closed orbits; zeta function