Stability of the braid types defined by the symplecticmorphisms preserving a link

Fix a suitable link on the disk. Recently, F. Morabito associates eachHamiltonian symplecticmorphism preserving the link to a braid type. Based onthis construction, Morabito defines a family of pseudometrics on the braidgroups by using the Hofer metric. In this paper, we show that two Hamiltoniansymplecticmorphisms define the same braid type provided that their Hoferdistance is sufficiently small. As a corollary, the pseudometric defined byMorabito is nondegenerate.