Square roots of Hamiltonian Diffeomorphisms

TL;DR

This paper proves that on any closed symplectic manifold, there exist arbitrarily small Hamiltonian diffeomorphisms that do not have a square root, revealing new insights into the structure of Hamiltonian diffeomorphisms.

Contribution

It demonstrates the existence of small Hamiltonian diffeomorphisms without square roots on all closed symplectic manifolds, a novel result in symplectic topology.

Findings

Existence of small Hamiltonian diffeomorphisms without square roots

Arbitrarily $C^\infty$-small examples on any closed symplectic manifold

New constraints on the algebraic structure of Hamiltonian diffeomorphisms

Abstract

In this article we prove that on any closed symplectic manifold there exists an arbitrarily $C^\infty$-small Hamiltonian diffeomorphism not admitting a square root.