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On the fixed-point proportion of self-similar groups

Jorge Fariña-Asategui, Santiago Radi

TL;DR

The paper shows that groups acting on regular rooted trees that are super strongly fractal have zero fixed-point proportion, extending a line of work that connects group actions with arithmetic dynamics. It combines Jones's martingale-based strategy with a novel ergodic perspective on self-similar groups to prove $\mathrm{FPP}(G)=0$ for SSF groups, and then applies this to iterated monodromy groups of dynamically exceptional polynomials, constructing explicit examples for all degrees $d\ge3$ with $\mathrm{FPP}(IMG(f))=0$. The results bridge automata/group-theoretic dynamics with arithmetic dynamics by showing that dynamically exceptional polynomials can yield IMG with vanishing fixed-point proportion, thereby supporting conjectures about density of primes dividing orbit terms. The work also clarifies the scope of the main theorem relative to contracting/finite-type groups, showing both the reach and the limitations of the SSF framework through concrete counterexamples and comparisons.

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].

On the fixed-point proportion of self-similar groups

TL;DR

The paper shows that groups acting on regular rooted trees that are super strongly fractal have zero fixed-point proportion, extending a line of work that connects group actions with arithmetic dynamics. It combines Jones's martingale-based strategy with a novel ergodic perspective on self-similar groups to prove for SSF groups, and then applies this to iterated monodromy groups of dynamically exceptional polynomials, constructing explicit examples for all degrees with . The results bridge automata/group-theoretic dynamics with arithmetic dynamics by showing that dynamically exceptional polynomials can yield IMG with vanishing fixed-point proportion, thereby supporting conjectures about density of primes dividing orbit terms. The work also clarifies the scope of the main theorem relative to contracting/finite-type groups, showing both the reach and the limitations of the SSF framework through concrete counterexamples and comparisons.

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].

Paper Structure

This paper contains 23 sections, 14 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. About general notation
  4. Groups acting on rooted trees
  5. Fractal properties of groups acting on trees
  6. Groups generated by automata
  7. Fixed point proportion
  8. Fixed-point proportion for super strongly fractal groups
  9. Fixed-point process and martingales
  10. Proof of \ref{['theorem: FPP SSF']}
  11. The fixed-point proportion of iterated monodromy groups of dynamically exceptional polynomials
  12. A primer on iterated monodromy groups
  13. Hyperbolic polynomials
  14. Construction of the iterated monodromy group of a polynomial using spiders
  15. Dynamically exceptional polynomials
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $K$ be any number field, $M_K^0$ the set of prime ideals in $\mathcal{O}_K$, $v_\mathfrak{p}$ the valuation in $\mathcal{O}_K$, $f$ any rational function in $K(x)$ of degree at least $2$ and $a_0 \in K$. Denote by $f^n$ the $n$-th iteration of $f$ and define the set namely, the set of prime ideals that divide at least one non-zero term in the orbit of $a_0$ by $f$. If $0$ does not appear infi

Figures (2)

  • Figure 1: CW complex of the automata group generated by the automorphisms $a = (1,1,1)(1\,2\,3)$ and $b = (a,a^{-1},b)$. This complex is not contractible and thus the automata group is not tree-like.
  • Figure 2: Venn diagram summarizing the comparison between \ref{['theorem: FPP SSF']} and \ref{['theorem: FPP contracting Jones']}.

Theorems & Definitions (19)

  • Theorem 1.1: see JonesManes2012
  • Theorem 1.2: see BridyJones2022, juul2014wreath and Self_similar_groups
  • Theorem 1
  • Theorem 1.4: see jones2012fixedpointfree
  • Theorem 2
  • Lemma 2.1
  • proof
  • Theorem 3.1: see Grimmett_Random_processes
  • Lemma 3.2: see jones_arborealsurvey
  • proof
  • ...and 9 more