Real cubic polynomials with a fixed point of multiplicity two

The aim of this paper is to study the dynamics of the real cubic polynomials that have a fixed point of multiplicity two. Such polynomials are conjugate to fa(x)=ax2(x−1)+xfa(x)=ax2(x−1)+x, a≠0a≠0. We will show that when a>0a>0 and x≠1x≠1, then |fan(x)| converges to 0 or ∞∞ and, if a<0a<0 and aa belongs to a special subset of the parameter space, then there is a closed invariant subset ΛaΛa of RR on which fafa is chaotic.