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

TL;DR

This paper constructs new examples of regular branch groups with positive Hausdorff dimension and fixed-point proportion, answering a longstanding question and expanding understanding of Galois groups in dynamical systems.

Contribution

It proves that certain iterated monodromy groups of unicritical polynomials are regular branch with positive Hausdorff dimension and fixed-point proportion, providing explicit computations and new examples.

Findings

Regular branch groups have positive Hausdorff dimension.

Constructed examples with positive fixed-point proportion.

First known examples outside binary rooted trees with these properties.

Abstract

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).