Flexibility and degeneracy around a theorem of Thurston

We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston’s asymmetric metric on Teichmüller space $\mathcal{T}(S_g)$ that remain geodesics after adding arbitrary $\epsilon$-Lipschitz noise to all but one Fenchel-Nielsen coordinate. Then, for all $2<n\le 3g-3$ we construct open sets in $\mathcal{T}(S_g{)}^n$ for which the limit cones of the corresponding representations in ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{R}{)}^n$ are cones over explicit finite-sided polyhedra. Each construction is as degenerate as possible and has applications to the basic structure and local non-rigidity of the involved objects.