Decay of geometry for a class of cubic polynomials

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.