Decay of geometry for a class of cubic polynomials

Abstract

In this paper we study a class of bimodal cubic polynomials satisfying two critical points belong to one 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 its twin principal nest decreases at least exponentially fast. As an application, we prove that they have no Cantor attractor