Thin monodromy and Lyapunov exponents, via Hodge theory

I will discuss a connection between monodromy groups of variations of Hodge structure and the global behavior of the associated period map. The large-scale information in the period map is contained in the Lyapunov exponents, which are invariants coming from dynamical systems. In some cases when the monodromy group is thin, i.e. infinite-index in the relevant arithmetic lattice, one can construct new geometric objects that cannot exist in the arithmetic case.