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

**Zeraoulia Elhadj and J. C. Sprott**
Department of Mathematics, University of Tébessa, (12002), Algeria

Department of Physics, University of Wisconsin, Madison, WI 53706,
USA

## Abstract

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)