Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials

Abstract: Let $f(x) \in K(x)$ be a quadratic polynomial where $K$ is a field of characteristic not equal to $2$. The associated arboreal Galois representation of the absolute Galois group of $K$ acts on a regular rooted binary tree. Boston and Jones conjectured that, for $f \in \mathbb{Z}[x]$, the image of this representation contains a dense set of settled elements. Roughly speaking, a cycle of an automorphism $τ$ of the tree is called stable if its length strictly increases at each subsequent level, and $τ$ is called settled if the proportion of vertices contained in stable cycles goes to $1$ as the level goes to infinity. In this article, we prove that the arithmetic iterated monodromy groups of postcritically finite quadratic polynomials in $K[x]$ with periodic postcritical orbits are densely settled. In the number field case, by a result of Benedetto--Ghioca--Juul--Tucker \cite{BGJT2025s}, it follows that for infinitely many $a \in K$, the associated arboreal Galois representations are densely settled. In particular, our results apply to the arithmetic IMG of the Basilica map $f(x)=x<sup>2-1$.

• The paper establishes that arithmetic iterated monodromy groups of quadratic PCF polynomials with periodic postcritical orbits are densely settled, confirming Boston and Jones' conjecture.
• It employs explicit construction of odometers and their normalizers to derive uniform estimates on the proportion of non-stable vertices at each level of the associated binary tree.
• The study provides an optimal classification with concrete numerical bounds on 2-adic valuations, paving the way for further research in arithmetic dynamics and arboreal Galois theory.

Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials

Introduction and Motivations

The investigation of arboreal Galois representations associated with quadratic polynomials aligns with foundational questions in arithmetic dynamics regarding the structure of fields generated by preimages under iteration. Boston and Jones previously conjectured that, for quadratic $f \in \mathbb{Z}[x]$, the corresponding arboreal Galois representations are densely settled—a measure-theoretic property for subgroups of the automorphism group of a binary tree. This paper resolves the conjecture for postcritically finite (PCF) quadratic polynomials with a periodic postcritical orbit, establishing dense settledness for their arithmetic iterated monodromy groups (IMGs).

Arboreal Galois Theory and Iterated Monodromy Groups

Given a quadratic $f \in K[x]$, the absolute Galois group $G_K$ acts on the tree of iterated preimages of a generic point $a \in K$. The resulting arboreal representation $\rho_{f,a}: G_K \to \Aut(T)$ encapsulates the Galois action on the $2$-ary rooted tree $T$ corresponding to $f$. The image, denoted $G_{f,a}$, is central to understanding field extensions generated by roots of iterates $f^n(x) - a$.

The arithmetic IMG arises naturally as a quotient of the étale fundamental group of $f \in K[x]$0, where $f \in K[x]$1 is the postcritical set. In the PCF case, this group is a profinite self-similar group capturing the monodromy of $f \in K[x]$2 over varying base points. The paper leverages Pink’s classification of these groups for periodic PCF quadratic polynomials.

Stable and Settled Elements: Definitions and Relevance

Boston and Jones introduced "settledness" as a fine-grained statistic on automorphisms of rooted trees, motivated by heuristics in dynamical factorization modulo primes. An automorphism is settled if almost all vertices at large depth are contained in stable cycles (cycles whose length doubles with each tree level). A subgroup is densely settled if its settled elements are dense in the profinite topology.

Prior work (Cortez-Lukina) showed densely settled IMGs for strictly preperiodic PCF quadratic $f \in K[x]$3, but the periodic case remained open. This paper resolves that gap and provides a strong structural analysis.

Structure Theory: Odometers, Normalizers, and Explicit Group Actions

Central to the analysis is the explicit construction of odometers (infinite order automorphisms acting as a single cycle at all levels) and their normalizers in $f \in K[x]$4. By leveraging self-similarity and recursively defined automorphisms (specifically, $f \in K[x]$5 as the standard odometer), the paper provides—

• A complete description of the normalizer of the closed subgroup generated by $f \in K[x]$6, showing it is the set $f \in K[x]$7, where $f \in K[x]$8 conjugates $f \in K[x]$9 to $G_K$0.
• Uniform estimates on the proportion of vertices not in stable cycles for elements in the normalizer, showing this proportion is always $G_K$1 at level $G_K$2.
• An optimal classification of settledness for elements in the normalizer in terms of 2-adic valuations.

Main Technical Results

The core theorems demonstrated are:

• The arithmetic IMG of any quadratic PCF $G_K$3 with a periodic postcritical orbit is densely settled.
• For any number field $G_K$4 and such an $G_K$5, for infinitely many $G_K$6, the specialized arboreal representation $G_K$7 is densely settled (a direct consequence of specialization theorems [BGJT2025s]).

The proof synthesizes Pink's explicit group computation with novel methods:

• The normalizer of the odometer in the arithmetic IMG is shown to capture all possible diagonal actions on the abelianization of the group.
• The notion of descendants (iterative sections of automorphisms under the wreath recursion) is refined, with stable blocks defined as elements all of whose descendants remain blocks with well-controlled structure.
• An inductive argument on the word-length in the natural generators (mimicking the tree depth) reduces settledness for arbitrary elements to the uniform estimates for odometer normalizers.

Concrete numerical bounds are given: for $G_K$8, any element $G_K$9 in the normalizer has at most $a \in K$0 non-stable vertices at depth $a \in K$1, where $a \in K$2. The recursive argument shows this is optimal.

Implications and Theoretical Consequences

By establishing dense settledness for the IMGs in the periodic PCF quadratic case, the paper solves a key conjecture and completes the known taxonomy of settledness for quadratic polynomials. The explicit nature of the arguments shows that the image of the Galois action on the tree not only has the right generic largeness (in analogy with open image theorems in arithmetic geometry), but in fact virtually all automorphisms in the arithmetic IMG display the maximal possible growth of stable cycles.

An important technical point is that the densely settled property lifts to finite index subgroups, so the results pass robustly to arboreal Galois groups arising from specializations.

Future Directions

This framework sets the stage for addressing several open problems:

• Extension to non-quadratic and non-uni/tricritical polynomials, where the self-similarity is more complicated and distinct odometer classes appear.
• Application to the Chebotarev density conjecture for dynamical fields, since settledness is intimately connected to the distribution of cycle types in reductions modulo primes.
• Investigation of exceptional behavior in the non-PCF case, where prior work has exhibited non-settled and even rigid representations.

The methods—especially the use of explicit normal forms, descendants and stable block induction—are expected to generalize to wider families of self-similar profinite groups and arithmetic monodromy groups of more general postcritically finite dynamical systems.

Conclusion

This paper establishes that the arithmetic iterated monodromy groups attached to quadratic PCF polynomials with periodic postcritical orbit are densely settled, fully resolving the conjecture of Boston and Jones in this case. The methods introduced, especially the new inductive classification and explicit normalizer control, represent a major technical consolidation for the dynamics and Galois theory of arboreal representations. The results pave the way for new developments in arithmetic dynamics, explicit Galois constructions, and the interplay of profinite group theory and number-theoretic dynamics.

Reference: "Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials" (2604.04524)