Branch iterated Galois groups with positive fixed-point proportion and positive Hausdorff dimension

In this article we prove that the arithmetic profinite iterated monodromy group of a post-critically infinite unicritical polynomial is regular branch (and so of positive Hausdorff dimension), and has positive fixed-point proportion when the degree is odd. The examples are instances of a bigger family of regular branch groups constructed in this article, whose fixed-point proportion can be computed explicitly and is positive in many cases. This gives the first examples outside the binary rooted tree where a level-transitive group has positive Hausdorff dimension and positive fixed-point proportion, answering in the negative a question of Jones (2008).