Applications of Square Roots of Diffeomorphisms

In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C ∞ -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C ∞ -small contactomorphism which is not “autonomous„. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) ∖ Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M