Appears in collection : 2022 - T2 - WS2 - Self-similarity of groups, trees and fractals

The Basilica Group is the iterated monodromy group associated with the Basilica map $z^2-1$. The Schur renormalization transformation describes the relation between the densities of states in various scales. For this group, it happens to be a rational map in two variables. We will describe the dynamical structure of this map that yields the laminar structure for the associated Green current (restricted to the real plane). It implies that the density of states for this group is a Cantor set of positive Lebesgue measure.

It is a joint work with E.Bedford, N.-B. Dang, and R.Grigorchuk.