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We consider the flows generated by generic gradients of Morse maps of aclosed connected manifold $M$ to a circle. To each such flow we associate aninvariant counting the closed orbits of the flow. Each closed orbit is countedwith the weight derived from its index and homotopy class. The resultinginvariant is called the eta function, and lies in a suitable quotient of theNovikov completion of the group ring of the fundamental group of $M$. Itsabelianization coincides with the logarithm of the twisted Lefschetz zetafunction of the flow. For $C^0$-generic gradients we obtain a formulaexpressing the eta function in terms of the torsion of a special homotopyequivalence between the Novikov complex of the gradient flow and the completedsimplicial chain complex of the universal cover of $M$.

Closed Orbits of Gradient Flows and Logarithms of Non-Abelian Witt Vectors