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

Considering a family of rational maps TnλTnλ relating to renormalization transformations, we prove that each Fatou component of TnλTnλ is a Hölder domain for odd integers n≥5n≥5 and some parameters λ∈(1,2)λ∈(1,2), but TnλTnλ 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(Tnλ)J(Tnλ), but the two endpoints of this interval belong to the boundary of some Fatou component of TnλTnλ.