Quasisymmetric geometry of Sierpiński carpet Julia sets
Volume 244 / 2019
Fundamenta Mathematicae 244 (2019), 73-107
MSC: Primary 37F45; Secondary 37F10.
DOI: 10.4064/fm494-12-2017
Published online: 7 September 2018
Abstract
In this paper, the main focus is on the Sierpiński carpet Julia sets of rational maps with non-recurrent critical points. We study the uniform quasicircle property of the peripheral circles, the relatively separated property of the peripheral circles and the locally porous property of these carpets. We also establish some quasisymmetric rigidities of these carpets, which generalizes the main results of Bonk–Lyubich–Merenkov (2016) to the postcritically infinite case. In the end we give a strategy to construct a class of postcritically infinite rational maps whose Julia sets are quasisymetrically equivalent to some round carpets.
