Fixed-Point Proportion in Geometric Iterated Galois Groups

• Fixed-point proportion is the asymptotic measure on a regular rooted tree that quantifies the proportion of elements fixing at least one end.
• A key classification links positive fixed-point proportions to Chebyshev polynomial cases, yielding precise values like 1/2 for odd degrees and 1/4 for even degrees.
• Martingale convergence, automata theory, and ergodic methods are employed to analyze group dynamics and derive significant arithmetic implications.

A fixed-point proportion (FPP) describes the asymptotic measure, in the sense of normalized Haar probability, of elements in a group acting on a regular rooted tree that fix at least one end (infinite ray). In the setting of geometric iterated Galois groups—profinite closed subgroups of tree automorphism groups arising from the action of absolute Galois groups on the preimage tree under iterations of a polynomial or rational function—this invariant has become foundational for understanding both group dynamics and arithmetic applications, particularly in arithmetic and arithmetic dynamics.

1. Fixed-Point Proportion in the Context of Rooted Trees

Let $T$ be the infinite regular $d$-ary rooted tree with vertex set identified with words over $X = \{1, \ldots, d\}$. Automorphisms $\mathrm{Aut}(T)$ preserve root and adjacency. For a closed subgroup $G \le \mathrm{Aut}(T)$, the level-$n$ quotient $T_n(G) = G / \mathrm{St}_G(n)$, where $\mathrm{St}_G(n)$ is the full stabilizer of all vertices up to distance $n$ from the root.

The level-$n$ fixed-point proportion is

$$\mathrm{FPP}_n(G) = \frac{\#\{g \in T_n(G): \exists v \in L_n,\, g(v) = v\}}{|T_n(G)|}$$

and the limiting fixed-point proportion is defined by

$$\mathrm{FPP}(G) = \lim_{n \to \infty} \mathrm{FPP}_n(G)$$

which is also the normalized Haar measure of the set of elements in $G$ that fix at least one end of $T$ (Fariña-Asategui et al., 28 Feb 2025, Jones, 2012, Fariña-Asategui et al., 22 Jan 2026).

2. Structure of Geometric Iterated Galois Groups and Their Proportions

Given $f \in K(x)$, $\deg f = d \ge 2$, and basepoint $a$, the tree $T$ formed by $f^{-n}(a)$, $n \ge 0$, supports an action of the absolute Galois group through the arboreal representation: $\rho_n: \Gal(K(a)^{\text{sep}}/K(a)) \to \Sym(f^{-n}(a)) \cong \Aut(T^{(n)})$ The geometric iterated Galois group is

$G_\infty^{\text{geom}}(K, f, a) = \rho(\Gal(K(t)^{\text{sep}} / K(t))) \leq \Aut(T)$

where $t$ is a transcendental. This group is typically fractal and self-similar, often level-transitive or branch, depending on dynamical properties of $f$ (Fariña-Asategui et al., 22 Jan 2026, Fariña-Asategui, 27 Jan 2026).

The relation between FPP and group structure is deep: when $G$ is super strongly fractal or "mixing" as a dynamical system, then $\mathrm{FPP}(G) = 0$. In contrast, positive FPP is possible precisely in circumstances connected to special "Euclidean orbifold" rational maps, such as Chebyshev polynomials (Fariña-Asategui et al., 28 Feb 2025, Fariña-Asategui et al., 22 Jan 2026).

3. Key Classification Theorems and Explicit Values

A central result is the complete classification for $f \in \mathbb{C}[x]$, $\deg f \ge 2$: $\FPP(G_\infty^{\text{geom}}(\mathbb{C}, f, t)) > 0 \iff f \text{ is linearly conjugate to } \pm T_d$ where $T_d$ is the Chebyshev polynomial. For these cases,

$\FPP(G) = \begin{cases} 1/2, & d \text{ odd} \ 1/4, & d \text{ even} \end{cases}$

These values are realized via dense infinite dihedral subgroups in the geometric iterated Galois group, with cycle analysis yielding the precise fixed-point probabilities (Fariña-Asategui et al., 22 Jan 2026). For all other complex polynomials, including all non-exceptional post-critically finite polynomials and dynamically exceptional cases with $\#E = 1$, the FPP is zero (Fariña-Asategui et al., 28 Feb 2025, Jones, 2012).

Explicit positive FPP also arises for certain families over $\mathbb{Q}$: for the polynomial $x^d + 1$ with odd $d \ge 3$, one computes

$\FPP(G_\infty) = \prod_{p\mid d} \frac{p-2}{p-1}$

Where $G_\infty$ is the closure of the geometric iterated Galois group of $x^d + 1$ over $\mathbb{Q}(t)$; the FPP is strictly positive for all odd $d$, and can be calculated in closed form using the structure of corresponding iterated wreath product groups (Radi, 2024, Fariña-Asategui, 27 Jan 2026).

4. Methods: Martingales, Automata, Ergodic Theory

A universal theme in proving results about fixed-point proportion is the use of stochastic processes on $G$, specifically: $$X_n(g) = \#\{ \text{vertices at level } n \text{ fixed by } g \}$$ This process is a nonnegative martingale under suitable transitivity conditions. The martingale convergence theorem dictates that $X_n \to X_\infty$ almost surely. Ergodic-theoretic methods are then used to show that, unless the group has special (non-mixing) structure, $X_n \to 0$ almost surely—implying vanishing FPP (Fariña-Asategui et al., 28 Feb 2025, Jones, 2012, Fariña-Asategui et al., 22 Jan 2026).

A secondary machinery is automata theory: many iterated monodromy groups can be described as the action of finite invertible automata (so-called kneading automata). Classification of elements via these automata enables effective enumeration of potential fixed-point elements and rigorous identification of exceptional cases (Jones, 2012).

Further, explicit construction of "bad monodromy" extensions (in which sufficiently many cosets in a quotient have all elements fixing some ray) is used to exhibit families with positive FPP in both the arithmetic and geometric Galois contexts (Fariña-Asategui, 27 Jan 2026, Radi, 2024).

5. Families with Positive Fixed-Point Proportion and Hausdorff Dimension

Recent work constructs infinite families of groups with explicit positive FPP and positive Hausdorff dimension. Given a transitive permutation group $P \le \Sym(d)$, the (profinite) iterated wreath product

$W_P = \varprojlim (P \wr P \wr \cdots)$

acts on $T_d$, and FPP can be computed as the largest fixed point in $[0,1]$ of an explicit polynomial $f_P(x)$. For $P = \mathrm{Aff}(d)$, as in $x^d + 1$, this gives

$\FPP(G_\infty) = \prod_{p | d} \frac{p-2}{p-1}$

A plausible implication is the existence, for each $d \ge 2$, of nontrivial subgroups of $\Aut(T_d)$ that are self-similar, level-transitive, of positive Hausdorff dimension, and possessing explicitly calculable, strictly positive FPP (Radi, 2024). These results generalize both the classical Chebyshev and deeply new families such as depth-2 pattern groups.

The Hausdorff dimension of such a group can be calculated as

$$\operatorname{HD}(G_\infty) = \frac{\log(d!)}{\log(d)}$$

demonstrating both their fractality and size within $\Aut(T_d)$ (Radi, 2024).

6. Consequences for Arithmetic Dynamics and Further Directions

The vanishing (or positivity) of FPP for geometric iterated Galois groups connects directly to longstanding questions in arithmetic dynamics. One consequence, via Chebotarev density, is that for non-exceptional cases the density of prime divisors in a polynomial orbit is zero. In exceptional (positive FPP) cases, there can be positive density of periodic points over finite fields, and a more intricate orbit-structure over number fields (Fariña-Asategui et al., 22 Jan 2026, Jones, 2012).

Extensions to the arithmetic side (full arboreal Galois representations) and to specializations at non-generic base points have been investigated in the context of self-similar group extensions and branching structures (Fariña-Asategui, 27 Jan 2026). Notably, the multidimensional parameter space of fixed-point processes and branch group constructions is now known to admit a far richer spectrum of positive FPP behavior than previously believed, including explicit counterexamples to older conjectures that ruled out large eventual count of fixed points (Fariña-Asategui, 27 Jan 2026).

The super strongly fractal and mixing group criteria now supply a unified framework for all currently known vanishing FPP results and their arithmetic implications, while recent generalizations address open problems regarding extension, specialization, and explicit calculation for entire new families of self-similar and branch groups.

7. Tables: Classification of FPP for Geometric Iterated Galois Groups

Polynomial $f(x)$	FPP Value	Key Paper
Chebyshev $T_d$, $d$ odd	$1/2$	(Fariña-Asategui et al., 22 Jan 2026)
Chebyshev $T_d$, $d$ even	$1/4$	(Fariña-Asategui et al., 22 Jan 2026)
$x^d + 1$, $d$ odd, $d \geq 3$	$\prod_{p | d} \frac{p-2}{p-1}$	(Radi, 2024, Fariña-Asategui, 27 Jan 2026)
Non-exceptional PCF	$0$	(Fariña-Asategui et al., 28 Feb 2025, Jones, 2012, Fariña-Asategui et al., 22 Jan 2026)
Dynamically exceptional, $\#E=1$	$0$	(Fariña-Asategui et al., 28 Feb 2025)

These classifications summarize the currently known landscape for the fixed-point proportion of geometric iterated Galois groups, as established by recent work (Fariña-Asategui et al., 28 Feb 2025, Fariña-Asategui et al., 22 Jan 2026, Radi, 2024, Fariña-Asategui, 27 Jan 2026, Jones, 2012). The periodic and preperiodic behavior, group-theoretic rigidity, and number-theoretic consequences of this invariant are areas of ongoing research.