Application of fragmentation norms to transported points by Hamiltonian isotopies

Morimichi Kawasaki, Ryuma Orita
RIMS Kôkyûroku, vol. 2098 (2018), pp. 57–59
Journal Page

Overview

#

We study conditions on the values of a Hamiltonian function $H$ that force its time-one map $\varphi_H$ to move some point by a definite distance. The aim is to find a condition that remains valid under small uniform perturbations of $H$, without controlling its derivatives.

On a closed surface of positive genus, we give such a condition using two non-contractible simple closed curves. If $H$ is sufficiently large on one curve and negative on the other throughout the isotopy, some point moves by more than a fixed distance $\varepsilon>0$. The proof combines a lower bound for a fragmentation norm with its local boundedness in the $C^0$ topology. We also formulate a related problem about Hamiltonian pseudo-rotations.

@article{kawasaki2018application,
  author  = {Kawasaki, Morimichi and Orita, Ryuma},
  title   = {{Application of fragmentation norms to transported points by Hamiltonian isotopies}},
  journal = {RIMS Kôkyûroku},
  volume  = {2098},
  pages   = {57--59},
  year    = {2018},
  url     = {http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/2098.html}
}