Graphicality, $C^0$ convergence, and the Calabi homomorphism
Bull. Korean Math. Soc. 2017 Vol. 54, No. 6, 2043-2051
Published online July 26, 2017
Printed November 30, 2017
Michael Usher
University of Georgia
Abstract : Consider a sequence of compactly supported Hamiltonian diffeomorphisms $\phi_k$ of an exact symplectic manifold, all of which are ``graphical'' in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the $\phi_k$ $C^0$-converge to the identity, then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the $\phi_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.
Keywords : $C^0$ Hamiltonian dynamics, Calabi homomorphism
MSC numbers : Primary 53D22
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