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 has the same property. The proof uses the approach of Rafe Jones based on martingales and a recent result of the first author on the dynamics of self‐similar groups.