Decay of Geometry for a Class of Cubic Polynomials

Abstract

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.

Keywords

Fibonacci / Cubic polynomial / Decay of geometry / Cantor attractor / 37E05 / 37F25

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Funding

National Natural Science Foundation of China(12301103)

Rights & permissions

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature