On bimodal polynomials with a non-hyperbolic fixed point

We consider the real polynomials of degree d + 1 with a fixed point of multiplicity d ≥ 2. Such polynomials are conjugate to fa,d(x) = axd(x − 1) + x, a ∈ R \ {0}. In this family, the point 0 is always a non-hyperbolic fixed point. We prove that for given d, d′, and a, where d and d′ are positive even numbers and a belongs to a special subset of R−, there is a′ < 0 such that fa,d is topologically conjugate to fa′ ,d′ . Then we extend the properties that we have studied in case d = 2 to this family for every even d > 2.