  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.
