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 f a ( x ) = a x 2 ( x − 1 ) + x , a ≠ 0 . We will show that when a > 0 and x ≠ 1 , then | f a n ( x ) | converges to 0 or ∞ and, if a 0 and a belongs to a special subset of the parameter space, then there is a closed invariant subset Λ a of R on which f a is chaotic.