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 is an element of 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.AMS Subject Classification: 37E05; 37E15;