Disjoint superheavy subsets and fragmentation norms

Morimichi Kawasaki, Ryuma Orita
Journal of Topology and Analysis, vol. 13 (2021), no. 2, pp. 443–468
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Overview

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A fragmentation norm measures how many local Hamiltonian motions are needed to express a given Hamiltonian diffeomorphism. More precisely, it counts the smallest number of factors whose supports lie in allowed open sets. This gives a way to study the geometry of the group $\mathrm{Ham}(M)$ through decompositions into local motions.

Using mutually disjoint subsets that are superheavy with respect to suitable invariants, we obtain lower bounds for fragmentation norms and, under appropriate spectral assumptions, construct bi-Lipschitz embeddings $I\colon \mathbb{R}^n\to\mathrm{Ham}(M)$. For a closed surface $\Sigma_g$ of genus $g\geq 2$ and the fragmentation norm associated with a contractible open set, such embeddings exist for every positive integer $n$. Restricting these embeddings to integer lattices answers a question of Brandenbursky.

@article{kawasaki2021disjoint,
  author        = {Kawasaki, Morimichi and Orita, Ryuma},
  title         = {{Disjoint superheavy subsets and fragmentation norms}},
  journal       = {Journal of Topology and Analysis},
  volume        = {13},
  number        = {2},
  pages         = {443--468},
  year          = {2021},
  doi           = {10.1142/S179352532050017X},
  eprint        = {1901.01647},
  archivePrefix = {arXiv}
}