[Paper Review] A dynamical perspective on shear-bend coordinates

Aaron Fenyes|arXiv (Cornell University)|Oct 20, 2015

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper introduces a dynamical framework for computing shear-bend coordinates of twisted SL₂ℂ local systems on compact surfaces, generalizing the abelianization procedure of Gaiotto, Hollands, Moore, and Neitzke beyond punctured surfaces. By leveraging dynamical systems tools—specifically, the behavior of stable/unstable lines under iterated linear maps—it provides a numerically tractable, intrinsic recipe for parameterizing these geometric structures, offering new insight into coordinate changes between different shear-bend parameterizations.

ABSTRACT

Twisted $\operatorname{SL}_2 \mathbb{C}$ local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface with punctures, Gaiotto, Hollands, Moore and Neitzke's "abelianization" process computes the shear-bend coordinates of a twisted $\operatorname{SL}_2 \mathbb{C}$ local system without reference to its hyperbolic geometry. Using tools from dynamics, we'll generalize abelianization to compact surfaces, leading to a dynamical recipe for the shear-bend parameterization. This recipe lends itself well to numerical approximation, and it may clarify the changes of coordinates that relate different shear parameterizations.

Motivation & Objective

• To extend the abelianization procedure for shear-bend coordinates—previously defined only for punctured surfaces—to compact surfaces using dynamical systems methods.
• To provide a geometrically intrinsic, numerically stable recipe for computing shear-bend coordinates without relying on hyperbolic geometry or pleated structures.
• To clarify the transformation rules between different shear-bend parameterizations by analyzing the dynamics of associated linear cocycles.
• To establish that the stable and unstable line fields on the unit tangent bundle are globally Lipschitz, ensuring regularity of the parameterization.

Proposed method

• Utilizes the dynamics of the linear holonomy action on the unit tangent bundle to define asymptotic stable and unstable line fields.
• Applies the concept of iterated linear maps to approximate the asymptotic behavior of geodesic trajectories, linking them to shear-bend parameters.
• Employs the triangle inequality and exponential decay estimates (e.g., d∠(Ex, Ex,mxy) ≲ e−2Kmxy) to bound angle distances between line fields.
• Constructs a globally Lipschitz map from the surface to the projective bundle of the bundle of lines, ensuring regularity of the parameterization.
• Generalizes the abelianization process from punctured surfaces to compact surfaces by replacing singularities with dynamical convergence.
• Uses the complex geometry of translation surfaces and meromorphic 1-forms to model puncture structures and their dynamical behavior.

Experimental results

Research questions

• RQ1 How can the abelianization procedure for shear-bend coordinates be extended from punctured surfaces to compact surfaces?
• RQ2 What dynamical properties of the linear holonomy action on the unit tangent bundle yield a well-defined parameterization of twisted SL₂ℂ local systems?
• RQ3 How do the stable and unstable line fields on the unit tangent bundle relate to the shear-bend coordinates?
• RQ4 What is the regularity of the map assigning to each point the asymptotic stable or unstable line, and how does it affect the parameterization?
• RQ5 How do changes of shear-bend coordinate systems relate to the dynamics of the associated linear cocycle?

Key findings

• The stable and unstable line fields on the unit tangent bundle are globally Lipschitz continuous, ensuring the parameterization is well-behaved and numerically stable.
• The angle distance between line fields decays exponentially with iteration steps, satisfying d∠(Ex, Ex,mxy) ≲ e−2Kmxy for large mxy.
• The shear-bend coordinates can be computed via a dynamical recipe based on iterated linear maps, independent of hyperbolic geometry or pleated structures.
• The method generalizes abelianization to compact surfaces, providing a new intrinsic way to compute these coordinates.
• The construction clarifies the transformation rules between different shear-bend parameterizations through the dynamics of the linear cocycle.
• The parameterization is surjective onto the non-elementary part of the twisted SL₂ℂ character variety for compact surfaces.