  In this paper we study a class of bimodal cubic polynomials for which its
critical points have the same $\omega$-limit set which is an invariant Cantor
set. These maps have generalized Fibonacci combinatorics in terms of
generalized renormalization on the twin principal nest. It is proved that such
maps possess `decay of geometry' in the sense that the scaling factor of the
twin principal nest decreases at least exponentially fast. As an application,
we prove that they have no Cantor attractor.
