Hamiltonian C0-continuity of Lagrangian capacity on the cotangent bundle

Abstract

Partially motivated by the study of topological Hamiltonian dynamics, we prove the following $C^0$-continuity of the Lagrangian capacity function ${\gamma }^{\mathrm{lag}}$:

$${\gamma }^{\mathrm{lag}}\left({\varphi }_H^1\right(o_N\left)\right):={\rho }^{\mathrm{lag}}(H;1)-{\rho }^{\mathrm{lag}}(H;[pt{]}^{\#})\to 0,$$ as ${\varphi }_H^1\to id$, provided the $H$’s satisfy $supp{X}_H\subset D^R\left(T^{\ast }N\right)\setminus o_B$ for some $R>0$ and a closed subset $B\subset N$ with nonempty interior. We also provide an estimate of the capacity in terms of the $C^0$-distance of $d_{C^0}({\varphi }_H^1,id)$ and the subset $B\subset N$ relative to $T^{\ast }N$.