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Disjoint superheavy subsets and fragmentation norms. (English) Zbl 1482.57027

Let \(M\) be a symplectic manifold and let \(\mathrm{Ham}(M)\) be its group of Hamiltonian diffeomorphisms. One can associate to each open cover \(\mathcal{U}\) of \(M\) a norm on \(\mathrm{Ham}(M)\) (called a fragmentation norm). The norm of a Hamiltonian diffeomorphism \(f\) is the infimum of all integers \(k\) such that \(f\) can be written as a product of \(k\) Hamiltonian diffeomorphisms \(f=f_{1}\cdots f_{k}\) where each \(f_{i}\) is generated by a time-dependent Hamiltonian compactly supported in some open set of \(\mathcal{U}\). These norms have been studied by many authors in the last 20 years, starting with the work of M. Entov and L. Polterovich [Int. Math. Res. Not. 2003, No. 30, 1635–1676 (2003; Zbl 1047.53055)].
In this work the authors introduce new lower bounds on these fragmentation norms and give sufficent conditions for the group of Hamiltonian diffeomorphisms of a symplectic manifold to contain the image of a bilipschitz injective homomorphism of a finite dimensional Euclidean space. These conditions rely on the notion of superheavy set, introduced by M. Entov and L. Polterovich [Compos. Math. 145, No. 3, 773–826 (2009; Zbl 1230.53080)].
Applications include the case where the symplectic manifold is the open unit ball of \(\mathbb{R}^{2n}\), \(\mathbb{S}^{2}\times \mathbb{S}^{2}\), \(\mathbb{CP}^{2}\) and closed surfaces and their direct products. Several open problems are mentioned at the end of the article.

MSC:

57R17 Symplectic and contact topology in high or arbitrary dimension
53D12 Lagrangian submanifolds; Maslov index
53D40 Symplectic aspects of Floer homology and cohomology
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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