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

2025/07/25 by Fernando Camacho-Cadena, Camacho-Cadena, Fernando
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2507.19191

openalex publication_date 2025/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The character variety \mathscrX(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∈π₁(S) induces a trace function tr_c\colon[ρ]↦ tr(ρ(c)) on \mathscrX(S,G). In this article, we study the Hamiltonian flows of trace functions associated to self-intersecting curves. We prove that when G=PSL(3,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 Θ-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.

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