  We show that if a closed manifold M admits an F-structure (possibly of rank
0) then its minimal entropy vanishes. In particular, this is the case if M
admits a non-trivial circle action. As a corollary we obtain that the
simplicial volume of a colsed manifold admitting an F-structure is zero. We
also show that if M admits an F-structure then it collapses with curvature
bounded from below. This is turn implies that M collapses with bounded scalar
curvature or, equivalently, its Yamabe invariant is non-negative. We show that
F-structures of rank zero appear rather frequently:every compact complex
elliptic surface admits one as well as any simply connected 5-manifold. We use
these results to study the minimal entropy problem. We show the following two
theorems: suppose M is obtained by taking connected sums of copies of CP^2
(with any orintation), S^2 \times S^2 and the K3 surface. Then M has zero
minimal entropy. Moreover, M admits a metric with zero topological entropy if
and only if M is diffeomorphic to S^4, CP^2, S^2 \times S^2, CP^2#CP^2 or
CP^2#(-CP^2). Finally, suppose that M is a closed simply connected 5-manifold.
Than M has zero minimal entropy. Moreover, M admits a metric with zero
topological entropy if and only if M is diffeomorphic to S^5, S^3 \times S^2,
the non-trivial S^3-bundle over S^2 or the Wu manifold SU(3)/SO(3).
