Applications of Square Roots of Diffeomorphisms

Yoshihiro Sugimoto

doi:10.3390/axioms8020043

Abstract

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.

Highlights

For any closed manifold M, the set of diffeomorphisms Diff( M ) forms a group and any one-parameter subgroup f : R → Diff( M ) can be written in the following form f (t) = exp(tX ).Here, X ∈ Γ( TM ) is a vector field and exp : Γ( TM) → Diff( M) is the time 1 flow of vector fields.From the inverse function theorem, one might expect that there exists an open neighborhood of the zero section U ⊂ Γ( TM) such that exp : U −→ Diff( M )is a diffeomorphism onto an open neighborhood of Id ∈ Diff( M )
One might expect that the set of “autonomous” diffeomorphisms
We prove that there exists an arbitrary C ∞ -small contactomorphism not admitting a square root

Summary

Introduction

From the inverse function theorem, one might expect that there exists an open neighborhood of the zero section U ⊂ Γ( TM) such that exp : U −→ Diff( M ). For a symplectic manifold ( M, ω ), the set of Hamiltonian diffeomorphisms Hamc ( M, ω ) contains “autonomous” subset Aut( M, ω ) which is defined by. There exists an arbitrary C ∞ -small Hamiltonian diffeomorphism in Hamc ( M, ω )\Aut( M, ω ). We prove that there exists an arbitrary C ∞ -small contactomorphism not admitting a square root. We prove that there exists an arbitrary C ∞ -small diffeomorphism in Diff0c ( M) not admitting a square root.

Main Result

Milnor’s Criterion

Proof of Theorem 1

Proof of Theorem 2