Hamiltonian systems and monotone twist mappings for braids

TL;DR

This paper demonstrates that any braid can be realized by a Hamiltonian system, linking braid theory with Hamiltonian dynamics, and shows that pseudo-Anosov braids correspond to pseudo-Anosov Poincaré maps.

Contribution

It extends Moser's technique to show that all braids, including pseudo-Anosov types, can be realized by Hamiltonian systems.

Findings

Any braid can be realized by a Hamiltonian system.

Pseudo-Anosov braids correspond to pseudo-Anosov Poincaré maps.

The method generalizes previous results on area-preserving maps.

Abstract

In 1986, Moser showed that for a given area-preserving map, there exists a Hamiltonian system that realizes it on the Poincar\'e section. Using his technique, we show that for any braid, there exists a Hamiltonian system whose orbits realize the given braid. In particular, when the braid is pseudo-Anosov, so is the Poincar\'e map of the corresponding Hamiltonian.