Flexibility of the adjoint action of the group of Hamiltonian diffeomorphisms

Abstract: The space of Hamiltonian diffeomorphisms has a structure of an infinite dimensional Frechet Lie group, with Lie algebra isomorphic to the space of normalized functions and adjoint action given by pull-backs. We show that this action is flexible: for a non-zero normalized function $f$, any other normalized function can be written as a sum of differences of elements in the orbit of $f$ generated by the adjoint action. Additionally, the number of elements in this sum is dominated from above by the $L_{\infty}$-norm of $f$. This result can be interpreted as an (bounded) infinitesimal version of the Banyaga's result on simplicity of $Ham(M,\omega)$. Moreover, it can be used to remove the $C^{\infty}$-continuity condition in the Buhovsky-Ostrover theorem on the uniqueness of Hofer's metric. This is joint work with Lev Buhovsky.

differential geometrydynamical systemsgeometric topologysymplectic geometryspectral theory

Audience: researchers in the topic

Geometry and Dynamics seminar

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