A Proof that S-Unimodal Maps are Collet-Eckmann Maps in a Specific Range of their Bifurcation Parameters

Generally, Collet-Eckmann maps require unimodality and multimodality. The inverse is not true. In this paper, we will prove that S-Unimodal maps are Collet-Eckmann maps in a specific range of their bifurcation parameters. The proof is based on the fact that the family of robustly chaotic unimodal maps known in the literature are all topologically conjugate to one another and the fact that if two S-unimodal maps of the interval are conjugate by a homeomorphism of the interval and if one of them is Collet-Eckmann, then so is the other one.

Ref: E. Zeraoulia and  J. C. Sprott, Acta Universitatis Apulensis 34, 51-55 (2013)