  We consider the break-up of invariant tori in Hamiltonian systems with two
degrees of freedom with a frequency which belongs to a cubic field. We define
and construct renormalization-group transformations in order to determine the
threshold of the break-up of these tori. A first transformation is defined from
the continued fraction expansion of the frequency, and a second one is defined
with a fixed frequency vector in a space of Hamiltonians with three degrees of
freedom.
