Continuous Hamiltonian dynamics and area-preserving homeomorphism group of \(D^2\) (Q2820980)

The main results of this paper relate to the question of continuously extending the Calabi invariant of area-preserving diffeomorphisms of the disk \(D^2\) to maps of less regularity. Let \(\mathrm{Diff}^\Omega(D^2,\partial D^2)\) denote the group of area-preserving diffeomorphisms of \(D^2\) that are supported in the interior. The Calabi invariant is a homomorphism NEWLINE\[NEWLINE\mathrm{Cal}: \mathrm{Diff}^\Omega(D^2,\partial D^2)\to \mathbb{R}.NEWLINE\]NEWLINE The existence of this homomorphism implies that the domain group is not simple.NEWLINENEWLINEIt is an open question whether the area-preserving homeomorphism group \(\mathrm{Homeo}^\Omega(D^2,\partial D^2)\) is simple. The obvious strategy of extending the Calabi invariant to homeomorphisms does not work, because for instance there is a sequence of area-preserving diffeomorphisms with Calabi invariant one that converge uniformly to the identity.NEWLINENEWLINEThe author conjectures that the Calabi invariant does admit a continuous extension to the group \(\mathrm{Hameo}(D^2,\partial D^2)\) of \textit{Hamiltonian homeomorphisms}. This group carries a metric that combines the \(C^0\) metric with the Hofer metric of the generating Hamiltonian.NEWLINENEWLINETo attack this conjecture, the author proves a version of the Alexander trick that holds in the context of Hamiltonian homeomorphisms. This reduces the first conjecture to a second conjecture that asserts the vanishing of a certain function called the \textit{basic phase function}. This function is a kind of Floer-theoretic spectral invariant.NEWLINENEWLINEThe author proves his conjecture on the vanishing of the basic phase function under the hypothesis that the homeomorphisms in question are \textit{graphical}, meaning that the graph of the homeomorphism has a one-to-one projection onto the diagonal.NEWLINENEWLINEIn an appendix, the author provides more material on the basic phase function, where he shows that it can be interpreted as giving a solution of the Hamilton-Jacobi equation.