Applications of square roots of diffeomorphisms (Q2306311)

Summary: In this paper, we prove that on any contact manifold \((M, \xi)\) there exists an arbitrary \(C^\infty\)-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary \(C^\infty\)-small contactomorphism which is not ``autonomous''. This paper is the first step to study the topology of \(Cont_0(M, \xi) \smallsetminus \mathrm{Aut}(M, \xi). As\) an application, we also prove a similar result for the diffeomorphism group \(\mathrm{Diff}(M)\) for any smooth manifold \(M\).