On the fixed-point proportion of self-similar groups

Abstract: We prove that super strongly fractal groups acting on regular rooted trees have null fixed-point proportion. In particular, we show that the fixed-point proportion of an infinite family of iterated monodromy groups of exceptional complex polynomials have the same property. The proof uses the approach of Rafe Jones in [15] based on martingales and a recent result of the first author on the dynamics of self-similar groups [6].