# Applications of square roots of diffeomorphisms

## TL;DR

This paper demonstrates that on contact manifolds, there are arbitrarily small contactomorphisms without square roots, highlighting the complexity of non-autonomous contactomorphisms and extending the result to diffeomorphism groups.

## Contribution

It proves the existence of small contactomorphisms without square roots on contact manifolds and extends this to diffeomorphism groups, advancing understanding of non-autonomous contact dynamics.

## Key findings

- Existence of small contactomorphisms without square roots
- Extension of results to diffeomorphism groups
- Implications for the topology of non-autonomous contactomorphisms

## Abstract

In this paper, we prove that on any contact manifold, 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 result is the first step to study the topology of non-autonomous contactomorphisms. As an application, we also prove a similar result for the diffeomorphism group for any smooth manifold.