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arxiv: 2506.09948 · v3 · submitted 2025-06-11 · 🧮 math.DS · math.AG

Periodic curves for general endomorphisms of mathbb Cmathbb P¹times mathbb Cmathbb P¹

Fedor Pakovich This is my paper

Pith reviewed 2026-05-19 09:44 UTC · model grok-4.3

classification 🧮 math.DS math.AG
keywords rational functionsfunctional decompositionperiodic curvesproduct endomorphismsconjugacyiteratescomplex dynamicsCP1 x CP1
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The pith

General rational functions have unique iterate decompositions, so their products admit non-horizontal-vertical periodic curves if and only if they are conjugate.

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

The paper proves that for a general rational function of degree at least two, the ways to factor its iterates into indecomposable pieces are all equivalent to the obvious one coming from the map itself. This rigidity result is applied to product endomorphisms on the product of two Riemann spheres. It yields a clean criterion: such an endomorphism has a periodic curve that is not a vertical or horizontal line precisely when the two component maps are conjugate. A sympathetic reader cares because the result turns an algebraic property of the maps into a dynamical statement about invariant curves in the product system.

Core claim

For a general rational function A of degree m greater than or equal to 2, any decomposition of its iterate A composed n times into a composition of indecomposable rational functions is equivalent to the decomposition of A composed n times itself. As an application, for a pair of general rational functions A1 and A2, the endomorphism of the product of two copies of the projective line given by sending each coordinate to its image under the respective map admits a periodic curve that is neither vertical nor horizontal if and only if A1 and A2 are conjugate.

What carries the argument

Uniqueness up to equivalence of the decomposition of iterates into indecomposable rational factors, for maps lying outside a proper algebraic subset of the space of all rational functions of fixed degree.

If this is right

  • If the two general maps are not conjugate then the product endomorphism has no periodic curves except the vertical and horizontal lines.
  • Conjugacy of the two maps is enough to guarantee the existence of at least one such non-trivial periodic curve.
  • The uniqueness of iterate decompositions holds for all but a lower-dimensional family of rational functions of each degree.
  • No unexpected factorizations appear in the iterates of a general rational function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conjugacy criterion may help decide the existence of other invariant sets in product dynamical systems built from rational maps.
  • One could check the result numerically by taking concrete non-conjugate pairs such as z squared and z cubed and searching for periodic curves in the product.
  • Similar rigidity arguments might classify periodic curves for endomorphisms of products of more than two spheres.

Load-bearing premise

The rational functions are general, lying outside a proper algebraic subset of the space of all maps of the given degree and thereby avoiding special cases such as decomposable or post-critically finite maps.

What would settle it

An explicit pair of non-conjugate general rational functions whose product endomorphism still possesses a periodic curve that is neither vertical nor horizontal would disprove the claim.

read the original abstract

We show that for a general rational function $A$ of degree $m \geq 2$, any decomposition of its iterate $A^{\circ n}$, $n \geq 1$, into a composition of indecomposable rational functions is equivalent to the decomposition $A^{\circ n}$ itself. As an application, we prove that if $(A_1, A_2)$ is a pair of general rational functions, then the endomorphism of $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$ given by $ (z_1, z_2) \mapsto (A_1(z_1), A_2(z_2)) $ admits a periodic curve that is neither a vertical nor a horizontal line if and only if $A_1$ and $A_2$ are conjugate.

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

2 major / 2 minor

Summary. The manuscript proves that for a general rational function A of degree m ≥ 2, any decomposition of the iterate A^{∘n} (n ≥ 1) into a composition of indecomposable rational functions is equivalent to the standard iterated decomposition of A itself. As an application, for a pair of general rational functions (A1, A2), the product endomorphism (z1, z2) ↦ (A1(z1), A2(z2)) on ℂℙ¹ × ℂℙ¹ admits a periodic curve that is neither vertical nor horizontal if and only if A1 and A2 are conjugate.

Significance. If the central claims hold, the result supplies a rigidity theorem for functional decompositions of iterates of rational maps, with direct consequences for the existence and classification of periodic subvarieties in product dynamical systems on surfaces. The algebraic-generality framework is a strength, as it yields a clean if-and-only-if statement in the application while systematically excluding post-critically finite or decomposable special cases.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1 (main decomposition theorem): The statement asserts the result for a single proper algebraic subset of the space of degree-m maps that works simultaneously for every n ≥ 1. The proof sketch treats the exceptional locus E_n for each fixed n separately via algebraicity of the decomposition condition; it is not shown that ⋃_n E_n is contained in a single proper closed subset (or that the loci stabilize for large n). This is load-bearing for the claim that a general A satisfies the property for all iterates.
  2. [§4, Theorem 4.2] §4, proof of the application (Theorem 4.2): The reduction from the existence of a non-horizontal/vertical periodic curve to a non-equivalent decomposition of some iterate relies on the main theorem holding uniformly in n. If the union issue in §3 is not resolved, the if-and-only-if direction for general (A1, A2) is not secured.
minor comments (2)
  1. [Introduction] The precise notion of 'equivalent decomposition' (presumably up to reordering and units in the function field) is used throughout but defined only in §2; a brief recall in the introduction would improve readability.
  2. [§2] Notation for the space of rational maps of degree m (e.g., the projective space of coefficients) is introduced in §2 but not consistently referenced when stating 'general' in the theorems; adding a parenthetical reminder would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to establish uniformity of the exceptional locus across all iterates. We respond to the major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] §3, Theorem 3.1 (main decomposition theorem): The statement asserts the result for a single proper algebraic subset of the space of degree-m maps that works simultaneously for every n ≥ 1. The proof sketch treats the exceptional locus E_n for each fixed n separately via algebraicity of the decomposition condition; it is not shown that ⋃_n E_n is contained in a single proper closed subset (or that the loci stabilize for large n). This is load-bearing for the claim that a general A satisfies the property for all iterates.

    Authors: We agree that the proof as written does not explicitly show that the union over n of the E_n lies in a single proper algebraic subset. The decomposition condition is algebraic in the coefficients of A, but the dependence on n must be controlled. We will revise the proof of Theorem 3.1 to establish that the exceptional loci stabilize for all n larger than a bound depending only on m (using the fact that any non-standard factorization of A^{∘n} for large n forces a non-standard factorization already at a fixed small iterate, via the structure of the monoid of rational functions). This yields a uniform proper algebraic exceptional set independent of n. revision: yes

  2. Referee: [§4, Theorem 4.2] §4, proof of the application (Theorem 4.2): The reduction from the existence of a non-horizontal/vertical periodic curve to a non-equivalent decomposition of some iterate relies on the main theorem holding uniformly in n. If the union issue in §3 is not resolved, the if-and-only-if direction for general (A1, A2) is not secured.

    Authors: The application in Theorem 4.2 indeed relies on the main decomposition result holding simultaneously for every iterate. Once the revised uniform statement of Theorem 3.1 is in place, the reduction from a non-trivial periodic curve to a non-equivalent decomposition of an iterate will be valid for general pairs (A1, A2), securing the if-and-only-if statement. We will update the proof of Theorem 4.2 to cite the strengthened version of the main theorem. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation relies on algebraic geometry and decomposition theory without self-referential reduction.

full rationale

The paper establishes a result on decompositions of iterates for general rational maps using standard tools from complex dynamics and algebraic geometry. The notion of 'general' is defined via avoidance of a proper algebraic subset in the parameter space of degree-m maps, and the claim is that this suffices to ensure the property holds for all iterates simultaneously. No quoted step reduces a prediction or theorem to a fitted parameter, self-citation chain, or definitional tautology. The application to periodic curves follows from the decomposition result without importing uniqueness via prior self-citation in a load-bearing way. The derivation is self-contained against external benchmarks in the field of rational function decomposition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard background in rational functions, iteration, and algebraic geometry over the complex numbers. No free parameters or invented entities are visible. The 'general' condition functions as a domain assumption excluding special cases.

axioms (2)
  • standard math The field of complex numbers and the Riemann sphere CP1 are the ambient spaces for rational functions and their iterates.
    Implicit in all statements about rational functions on CP1.
  • domain assumption Rational functions admit decompositions into indecomposable factors under composition.
    Used in the main theorem on uniqueness of such decompositions.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

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