Hamiltonian systems and monotone twist mappings for braids

TL;DR

This work shows that any braid on $n$ strands can be realized as the orbit structure of a Hamiltonian flow on an annulus via Moser's monotone twist framework, with a Hamiltonian $H$ whose time-1 map matches a given braid. For a $C^ ablafty$ monotone twist map $\varphi$, there exists a $C^{\infty}$ Hamiltonian $H$ with $H(t+1,x,y)=H(t,x,y)$ and $H_{yy}>0$ realizing $\varphi$ as its flow; the authors extend this to braids by constructing explicit generator Hamiltonians, starting from trivial braids and the elementary generator $\sigma_1$, using generating functions with carefully chosen rotation angles $\theta=\pm \frac{\pi}{2n}$ and a periodic perturbation to satisfy the twist condition $g_{xX}<0$. By composing and smoothing these generators, they build Hamiltonians whose flows realize concatenated braids, and they show that pseudo-Anosov braids yield pseudo-Anosov Poincaré maps, linking braid theory to Hamiltonian dynamics. This establishes a concrete method to realize braids topologically within Hamiltonian systems, with potential implications for understanding complex braid dynamics and their Poincaré maps.

Abstract

In 1986, Moser showed that for a given area-preserving map, there exists a Hamiltonian system that realizes it on the Poincaré 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 Poincaré map of the corresponding Hamiltonian.

Figures (1)

• Figure 1: Geometric $3$-braids

Theorems & Definitions (8)

• Definition 1.1: Monotone twist mappings
• Theorem 1.2: Moser86
• Theorem 1.3: Main Theorem $1$
• Corollary 1.4: Main Theorem $2$
• Remark 2.1
• Proposition 2.2
• Remark 2.3
• proof : Proof of Theorem \ref{['theorem:main']}