×

Continuous Hamiltonian dynamics and area-preserving homeomorphism group of \(D^2\). (English) Zbl 1356.53077

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 \[ \mathrm{Cal}: \mathrm{Diff}^\Omega(D^2,\partial D^2)\to \mathbb{R}. \] The existence of this homomorphism implies that the domain group is not simple.
It 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.
The author conjectures that the Calabi invariant does admit a continuous extension to the group \(\mathrm{Hameo}(D^2,\partial D^2)\) of Hamiltonian homeomorphisms. This group carries a metric that combines the \(C^0\) metric with the Hofer metric of the generating Hamiltonian.
To 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 basic phase function. This function is a kind of Floer-theoretic spectral invariant.
The author proves his conjecture on the vanishing of the basic phase function under the hypothesis that the homeomorphisms in question are graphical, meaning that the graph of the homeomorphism has a one-to-one projection onto the diagonal.
In 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.

MSC:

53D05 Symplectic manifolds (general theory)
53D35 Global theory of symplectic and contact manifolds
53D40 Symplectic aspects of Floer homology and cohomology
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
PDFBibTeX XMLCite
Full Text: DOI arXiv Link