Abstract
It is proved that a compact smooth manifold admits a selfdiffeomorphism without periodic points if and only if its Euler characteristic is zero. When the manifold has dimension $\ne 3$ it is shown that such a diffeomorphism exists which is also volume preserving. The proof of this latter result uses a result of Gromov concerning the existence of nonsingular divergence-free vector fields, so an alternate proof of Gromovâs result is sketched.
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