Twist-bulge derivatives and deformations of convex real projective structures on surfaces

A basic example of such a convex $\mathbb{RP}^2$ structure on a surface $S$ is the one associated to a hyperbolic structure on $S$, and in this special case Wolpert proved formulas for computing the Lie derivatives $t_{\alpha}l_{\beta}$ and $t_{\gamma}t_{\alpha}l_{\beta}$, where $t_{\alpha}$ is the Fenchel-Nielsen twist vector field associated to the twist along a geodesic $\alpha$, and $l_{*}$ is the hyperbolic geodesic length function.

In this dissertation, we extend Wolpert's calculation of $t_{\alpha}l_{\beta}$ and $t_{\gamma}t_{\alpha}l_{\beta}$ in the hyperbolic setting to the case of convex real projective surfaces; in particular, our $t_{\alpha}$ is the twist-bulge vector field along geodesic $\alpha$ coming from the parametrization of the deformation space of convex $\mathbb{RP}^2$ structures on a surface due to Goldman, and our geodesic length function $l_{*}$ is in terms of a generalized cross-ratio in the sense of Labourie.