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arxiv: 2403.12165 · v3 · submitted 2024-03-18 · 🧮 math.NT · math.DS· math.PR

Non-Martingale Fixed-Point Processes for Iterated Monodromy Groups

Jianfei He , Zheng Zhu This is my paper

Pith reviewed 2026-05-24 03:07 UTC · model grok-4.3

classification 🧮 math.NT math.DSmath.PR
keywords rational functionsfixed-point processesmartingalesarboreal Galois representationspostcritically finite mapsmonodromy groupsdynamical systemscritical orbits
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The pith

Rational functions of degree at least 2 exist whose fixed-point processes fail to be martingales.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit families of rational functions from the projective line to itself over perfect fields where the associated fixed-point processes are not martingales. It supplies geometric conditions on critical orbits that force the process to be a martingale for finite generically etale self-maps of normal varieties in projective space. These constructions answer a prior question on whether non-martingale behavior can appear in arboreal Galois representations and extend the criteria to higher-dimensional dynamical systems. The work also exhibits infinitely many postcritically finite maps with non-martingale processes and proves that the fixed-point proportion still vanishes at a quantifiable rate even when the martingale property fails.

Core claim

We construct families of rational functions f from the projective line over a perfect field k to itself of degree d at least 2 whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety X in projective space over the algebraic closure and finite generically etale self-map f on X, geometric conditions on the critical orbits guarantee that the fixed-point process is a martingale. Infinitely many postcritically finite maps admit non-martingale fixed-point processes, the group-theoretic obstructions in the genus-zero case are characterized, and the fixed-point proportion vanishes with quantifiable convergence rate despite the failure of the martingale law.

What carries the argument

The fixed-point process attached to the iterated monodromy group of the rational map f.

Load-bearing premise

The stated geometric conditions on the critical orbits are sufficient to force the fixed-point process to satisfy the martingale property.

What would settle it

An explicit rational function of degree 2 over a perfect field together with a direct computation of the second-moment increment of its fixed-point process that deviates from the martingale increment.

Figures

Figures reproduced from arXiv: 2403.12165 by Jianfei He, Zheng Zhu.

Figure 1
Figure 1. Figure 1: Ramification Portraits of g ◦ ψ. we get a 1-parameter family of degree 4 rational functions φ = r ◦ f = ax4 − ax2 + a + b ax4 − ax2 − a 2 − b = g ◦ h, where g = r ◦ g1 = ax2 + ax + a + b ax2 + ax − a 2 − b . In particular, by fixing the critical point 0 (let a + b = 0), we get a PCF rational function ψ = 2x 4 − 2x 2 2x 4 − 2x 2 + 1 . In fact, we can get infinitely many PCF rational functions by controlling… view at source ↗
Figure 2
Figure 2. Figure 2: We get a level 4 tree by decomposing φ as g ◦ h. we see that φ˜ : C → P 1 k is a finite separable morphism of smooth projective curves of genus 0, where K1 = k(C) is the function field of C. Now we show that its fixed-point process is not a martingale. By the nature of composition, we decompose our 4-ary tree of level two into a binary tree of level 4 (see figure 2). By computing the ramification portrait … view at source ↗
Figure 3
Figure 3. Figure 3: Tree of cosets Before we state the theorem, we first give an example to show that [17, Theorem 3.1] cannot be generalized to higher dimensions directly. Example. Let φ : P 1 k × P 1 k → P 1 k × P 1 k be a split morphism over an algebraically closed field k given by φ(x, y) = (φ1(x), φ2(y)), where φ1 and φ2 are disjoint rational functions from P 1 to itself. This induces a field extension M of k(t, u) gener… view at source ↗
read the original abstract

We construct families of rational functions $f \colon \bP^1_k \to \bP^1_k$ of degree $d \geq 2$ over a perfect field $k$ whose associated fixed-point processes fail to be martingales. Conversely, for any normal variety $X \subset \bP^N_{\overline{k}}$ and a finite, generically \'etale morphism $f \colon X \to X$, we establish geometric conditions on the critical orbits of $f$ that guarantee the fixed-point process is a martingale. Our constructions answer a question of Bridy, Jones, Kelsey, and Lodge \cite{iterated} regarding the existence of non-martingale behaviour in arboreal Galois representations, and extend their martingale criteria to higher-dimensional dynamical systems. In particular, we exhibit infinitely many postcritically finite maps with non-martingale fixed-point processes and characterize the group-theoretic obstructions to the martingale property in the genus-zero case. Furthermore, we prove that despite the failure of the martingale property, the fixed-point proportion still vanishes with a quantifiable convergence rate.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper constructs families of rational functions f: P^1_k → P^1_k of degree d ≥ 2 over a perfect field k whose fixed-point processes fail to be martingales, including infinitely many postcritically finite examples. Conversely, it gives geometric conditions on the critical orbits of a finite generically étale self-map f: X → X of a normal variety X ⊂ P^N that guarantee the fixed-point process is a martingale. The work answers a question of Bridy–Jones–Kelsey–Lodge on non-martingale behavior in arboreal Galois representations, extends genus-zero criteria to higher dimensions, characterizes group-theoretic obstructions in genus zero, and proves that the fixed-point proportion vanishes at a quantifiable rate even when the martingale property fails.

Significance. If the constructions and conditions are correct, the results are significant for arithmetic dynamics: they supply explicit counterexamples to martingale behavior (resolving an existence question) and sufficient geometric criteria that apply beyond curves. The extension to normal varieties, the PCF examples, and the independent convergence-rate statement are concrete advances. The absence of free parameters or post-hoc fitting in the stated claims strengthens the contribution.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'characterize the group-theoretic obstructions to the martingale property in the genus-zero case' is stated without indicating the precise form of the characterization (e.g., a subgroup index or ramification condition); a one-sentence pointer to the relevant theorem would improve readability.
  2. The manuscript should confirm that the quantifiable convergence rate for the fixed-point proportion is derived without assuming the martingale property (as asserted in the abstract); if this appears in a dedicated section, a cross-reference would help.
  3. Ensure the bibliography entry for Bridy–Jones–Kelsey–Lodge is complete and that any new notation for critical orbits or fixed-point processes is defined before first use in the main text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance for arithmetic dynamics, and recommendation of minor revision. No major comments appear in the report, so we have no point-by-point responses to offer. We will address any minor issues identified during the revision process.

Circularity Check

0 steps flagged

No significant circularity; constructions and conditions are independent theorems

full rationale

The paper states two main results as theorems: explicit families of degree-d rational maps over perfect fields with non-martingale fixed-point processes (including infinitely many postcritically finite examples), and sufficient geometric conditions on critical orbits guaranteeing the martingale property for finite generically étale self-maps of normal varieties. These are presented as direct answers to an external question from Bridy–Jones–Kelsey–Lodge and an extension of their genus-zero criteria. No equations reduce a claimed prediction to a fitted parameter by construction, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness result is smuggled via prior author work. The derivation chain consists of independent geometric and group-theoretic arguments that do not collapse to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, ad-hoc axioms, or invented entities; the work relies on standard facts from algebraic geometry, Galois theory, and probability that are not introduced in the paper itself.

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Reference graph

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