On the dynamics of a family of non-Collet–Eckmann rational maps

Considering a family of rational maps T n λ relating to renormalization transformations, we prove that each Fatou component of T n λ is a Hölder domain for odd integers n ≥ 5 and some parameters λ ∈ ( 1 , 2 ) , but T n λ is not a Collet–Eckmann map even if there is only one critical point in its Julia set. Furthermore, we show that there exists a buried open interval in the Julia set J ( T n λ ) , but the two endpoints of this interval belong to the boundary of some Fatou component of T n λ .