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