  We study the action of the mapping class group on the subspace of de Rham
classes in the degree-two bounded cohomology of a hyperbolic surface. In
particular, we show that the only fixed nontrivial finite-dimensional subspace
is the one generated by the Euler class. As a consequence, we get that the
action of the mapping class group on the space of de Rham quasimorphisms has no
fixed points.
