Two extensions of Thurston's spectral theorem for surface diffeomorphisms

Thurston obtained a classification of individual surface homeomorphisms via the dynamics of the corresponding mapping class elements on Teichm\"uller space. In this paper we present certain extended versions of this, first, to random products of homeomorphisms and second, to holomorphic self-maps of Teichm\"uller spaces.Comment: 11 page

Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular Embeddings

If the monodromy representation of a VHS over a hyperbolic curve stabilizes a rank two subspace, there is a single non-negative Lyapunov exponent associated with it. We derive an explicit formula using only the representation in the case when the monodromy is discrete.Comment: 22 pages, 4 figures; accepted version to be published in Ann. Inst. Fourier (Grenoble