Abstract
Let Q denote a quadratic polynomial and A∞ the super-attracting basin of Q at the point ∞ on the Riemann sphere [Formula: see text]. There exists a unique Riemann mapping Φ from the open disk [Formula: see text] onto A∞ such that Φ(∞)=∞, Φ'(∞)=1 and Φ-1 conjugates Q:A∞→A∞ to the squaring map S:D→D:z↦z2. In this paper, we show if Q is real and infinitely renormalizable of bounded type then the continuous extension of Φ to the closed disk cannot have any Hölder continuity on the boundary circle [Formula: see text].
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