Self-intersecting curves on a pair of pants and periodic orbits of Hamiltonian flows

TL;DR

This paper investigates Hamiltonian flows of trace functions on character varieties for a pair of pants, showing that certain flows are periodic with unique fixed points, using explicit coordinate computations.

Contribution

It demonstrates that Hamiltonian flows of specific trace functions on the pair of pants are periodic with unique fixed points, especially for self-intersecting curves, using explicit Fock-Goncharov coordinates.

Findings

Hamiltonian flows of figure-eight trace functions are periodic with a unique fixed point.

Results extend to the trace of the $ heta$-web and other self-intersecting curves.

Explicit coordinate computations underpin the proofs.

Abstract

The character variety $\mathscr{X}(S,G)$ associated to an oriented compact surface $S$ with boundary and a real reductive Lie group $G$ admits a Poisson structure and is foliated by symplectic leaves. When $G$ is a matrix group, any closed curve $c\in\pi_1(S)$ induces a trace function $\mathsf{tr}_c\colon[\rho]\mapsto \mathsf{tr}(\rho(c))$ on $\mathscr{X}(S,G)$. In this article, we study the Hamiltonian flows of trace functions associated to self-intersecting curves. We prove that when $G=\mathsf{PSL}(3,\mathbb{R})$ and $S$ is the pair of pants, every orbit of the Hamiltonian flow of the trace of a figure eight curve on $S$ is periodic and has a unique fixed point. The proof uses explicit computations in Fock-Goncharov coordinates. As an application, we prove a similar statement for the trace of the $\Theta$-web. Finally, we focus on the symplectic leaf corresponding to the unipotent locus, and derive similar results for two more self-intersecting curves: the commutator, and a curve going $k$ times around a boundary component.