Origamis are translation surfaces obtained by gluing finitely many unit squares and provide an
easy access to Teichmüller curves. In particular their monodromy represenation can be
explicitely determined. A general principle for the decomposition of this represenation is
exhibited and applied to examples. Closely connected to it is a dynamical cocycle on the
Teichmüller curve. It is shown that its Lyapunov exponents otherwise inaccessible can be
computed for a subrepresentation of rank two.
