  We develop a class of uniformizations for certain weight 3 variations of
Hodge structure (VHS). The analytic properties of the VHS are used to establish
a conjecture of Eskin, Kontsevich, M\"oller, and Zorich on Lyapunov exponents.
Additionally, we prove that the monodromy representations are log-Anosov, a
dynamical property that has a number of global consequences for the VHS. We
establish a strong Torelli theorem for the VHS and describe appropriate domains
of discontinuity.
  Additionally, we classify the hypergeometric differential equations that
satisfy our assumptions. We obtain several multi-parameter families of
equations, which include the mirror quintic as well as the six other thin cases
of Doran--Morgan and Brav--Thomas.
