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

Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents