arXiv:2610.31961v1 Announce Type: new Abstract: In this paper, we study the iteration over the real and complex loci of a specific rational map, found by Grigorchuk and Zuk. This two dimensional map arises as the renormalization map which yields the density of states measure associated with the iterated monodromy group of the Basilica Julia set. It admits two natural complex currents, and we construct their real analogs and show that they are laminar. Finally, this yields some consequences on the properties of the density of states measures and its support.
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